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^ 




Stables and rules 

FOR FACILITATING THE CALCULATION 






EARTHWORK, 
LAND, CURVES, 

DISTANCES, AND GRADIENTS, 

REQUIRED IN THE FORMATION 

OF 

RAILWAYS, 

ROADS, AND CANALS: 

ALSO, 

ESSAYS 

ON THE PRISMOIDAL FORMULA, 

AND 

ON THE POWER REQUIRED UPON INCLINED PLANES. 




BY J. B. HUNTINGTON, 

A880CIATB OP THB INSTITUTION OF CIVU< BVaiNBBRS. 



>■ 



*i: 



LONDON: 
*:? PUBLISHED FOR THE AUTHOR. 

BY JOHN WEALE, HIGH HOLBORN. 






[entered at stationers' hall«] 

1846. 
Vriot Twenty-four Shillingt. 



•v 



M 



*..t 



TfUtportatlon 
Library 

TA 



LONDON: 

PRINTED BY JOSEPH ROGERSOM^ 

94, VOBVOLK 8T&BBT, 8TBAKD. 



t 



Libra ly 



PREFACE. 

• 

The great advancement made of late years in the sci- 
ence of civil engineering, and the increased number of 
works promoted by public and private interests, in the 
formation of roads, canals, and railways, call for a 
more extended view of the theory of the construction 
and measurement of such works than has hitherto been 
given to the world, both for the better instruction of 
pupils, and for the facihtation of the labours of practical 
men. With the above object in view I have prepared 
the following work, especially adapted to the " measure- 
ment" of railways, and comprising in its extent a 
greater compilation of Tables and Essays, on the impor- 
tant features of this subject, than has hitherto been 
introduced. 

During the process of calculating the cubic quantity 
of earthwork requisite to form a given embankment, 
or to be taken from a given cutting, I have found the 
usual methods proposed in various authors and geome- 
tric writers more or less tedious in their application ; 
and in the more improved works furnished to the public. 



II 



the extent of the tables has not been sufficient, in some 
instances, to enable them to be serviceable. 

Calculations of this description are frequently required 
on a very short notice ; and then, on account of the 
multiplicity of figures employed, an easy and accurate 
method is an object, not only because time is saved, 
but amongst so many evolutions of numeration, a ready 
method would greatly tend to prevent errors to which 
a lengthened process is liable ; and I hope that, on a 
fair investigation into the merits of my manner of treat- 
ing this subject, practical men, both engineers and 
contractors for works, may judge favourably of the 
utility of my labours. 

The portion of the work devoted to the calculation 
of earthwork contains " Tables for all Bases, between 
Base 20 and Base 36 inclusive, adapted to ten different 
Slopes." The tables are followed by an " Essay on 
the Prismoidal Formula," and a full " Explanation of 
the use of the Tables," with a digest of all the " Rules 
derived from the Essay." "Tables for calculating 
Areas of Land" and " Areas of Slopes" are next given, 
accompanied by an " Explanation." Then follows a 
set of " Tables for calculating Offsets to Curves," and 
an "Explanation" of its use, together with a short 
"Table to find the Radii of Curves," and "Rules 
relating to Curves." Next to this comes a "Table 
of Distances," and then a "Table of Gradients," 
with the necessary " Explanation and forms of use." 



Ill 



I conclude the work with an " Essay on the Power 
required on IncKnes/* in which all the circumstances of 
Friction, Atmospheric resistance. Gravity, Loss of power 
round Curves and by the action of the Wind, are all 
separately considered. 

In order to give a general notion of the various 
conditions which the Earthwork of a Railway exhibit, 
I have prefixed a Frontispiece, which, in a bird's-eye 
view, pourtrays a cutting ^ with the eflFect produced by 
^ding ground, and a hench formed on the highest side, 
at a certain height, extending along the cutting, parallel 
with the base, till it meets the surface of the ground ; 
also an imfinished embankment, with the foreground 
iflustrating the fencing off of the requisite land, and 
other minor points of appearance. 

The great object I have endeavoured to keep in view 
has been utility , especially to students in Civil Engi- 
neering. I have, therefore, avoided all subjects the 
investigation of which is merely iogenious without 
practical aim. Entire novelty of matter is hardly to be 
expected in a work of this nature ; but I humbly hope 
that, in general, the reader will not feel disappointed in 
this respect. 

In choosing the t3rpe of this work, especially for the 
tables, my object was to obtain a bold, clear figure, 
which should admit of being easily read off; in which 
important desideratum, I trust, the reader will admit 
that I have succeeded. 



All the calculations will be founded on theories 
derived from or confirmed by practical observations ; 
and the data, where assumed in illustration of the rules, 
will be such as may be considered an average, but 
which, owing to the many complex and inappreciable 
difficulties which invest discussions such as are con- 
tained in this volume, may be varied by the judgment 
of the reader. The work is so arranged, that the por- 
tion relating to the Calculation of Earthwork can be 
separated, and two volumes can be made up instead of 
one, to suit the convenience of the reader. I request 
indulgence for such irregularities and errata as may 
occur in the composition of the essays; and I will gladly 
acknowledge any communications and corrections, and 
insert them in subsequent volumes. 

J. B. HUNTINGTON. 
Wamtead, Eaaex, 
March \8t, 1846. 



CONTENTS. 



EARTHWORK. 



Page. 



(S 



Tables for calculating Earthwork, from Base 20 to Base 
36, with Slopes i to 1, ^ to 1^ | to 1, 1 to 1, li to 1, li to 1, 

lito2to l,2^to l,3to 1 1 

Tables for Base 20 1 

11 

21 

31 

41 

51 

61 

71 

81 

91 

101 

..' Ill 

121 

131 

141 

151 

161 

m 

173 

173 

l74 



21 

22 

23 

24 

25 

26 

27 

28 

29 

30 

31 ' 

32 

33 

34 

35 

36 

Table of Bases only 

Prismoidal Formula (Essay on the) 

Prismoid (Description of) 

Section (Description of) • 

Rule for computing Prismoid, as given in Books of Geometry... . 176 

Proof of the above rule by the method of Fluxions 176 

Proof of the Rule by Geometric demonstration . .• 178 

Prismoidal Formula compared with the method of computing by 
McanHeights 179 



€t 



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tt 



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VI CONTENTS. 

Page, 
Prismoidal Formula compared with the method of computing by 

Mean Areas 180 

Examples in illustration of the above Rules 181 

Cross Sections illustrated 184 

Areas of Prismoid, with imiform Sloping Cross Sections (Rule for 

finding) 187 

Ditto, Rule compared with the Rule for Level Ground 188 

Cubic Contents of Prismoids, with uniform Sloping Cross Sections 

(Rule for finding) '; *188 

Ditto^Rule compared with the Rule for Level Ground 189 

Areas and Cubic Contents of Prismoids, with ^'ariable Cross 

Sections (Rules for finding) 190 

Areas of Prismoids, with curving Cross Sections (Rule for finding) 192 

Rule for Prismoids with two dififerent Slopes 193 

Rule for finding the Cube due to an alteration of Level of the 

Base Line 194 

Rule for the Area of the Prismoid when Benches are introduced. . 195 
Rule for Area of the Prismoid when the Widths at the Surface are 

given instead of the Heights • 196 

Rule for the Cubic Contents of the Prismoid, as above 197 

Rule for the Area of the Prismoid, with uniform Sloping Cross 

Sections . « . » 198 

Rule for the Cubic Contents of the Prismoid, as above 199 

Practical Rules derived from the Essay relating to the 

Prismoidal Formula (Compilation of) 199 

Explanation of the use of the Tables 203 

Examples by MacNeil's Tables of Earthwork 204 

][lule for forming MacNeil's Tables of Earthwork 204 

Rule for forming the Tables of Earthwork in this Work 205 

Rule to calculate the Cube of a Prismoid by the Tables 205 

Form of arranging a series of Calculations for a Railway, &c.. . . . 207 

Rule to calculate the Area of a Prismoid by the Tables 208 

Role to calculate the Area of the Sides of a Prismoid by the 

Tables. 208 

Rule to calculate the Cubic Contents of a Prismoid when the Base 

is altered 209 

Rule to calculate the Area of a Prismoid when the Base is altered 209 



CONTENTS. Vll 

Page. 

Explanation of a Scale for calculating Earthwork 210 

Application of the above Scale 211 

Rule for constructing the above Scale. . 211 

LAND. 

Tables for calculating the Areas of Land 213 

Tables for calculating the Areas of Slopes 218 

Explanation of the Tables for calculating Areas of Land 223 

Explanation of the Tables for calculating the Areas of Slopes .... 224 

CURVES. 

Tables of Offsets to Curves, for a Radius = 1000 225 

Explanation of the Tables of Offsets to Curves 228 

Rule for calculating Offisets 229 

Rule for the use of the Tables of Offsets 229 

Examples of the preceding Rule 230 

General Rules for calculating Tangents, Offsets, Chords, and 

Radii of Curves 231 

Rule for calculating the Radius when the Tangent and 

Angle are given 232 

DISTANCES. 

• 

Explanation of the Table of Distances 233 

Tables op Distances in Miles, Chains, Yards, and Feet 234 

Tables of Distances in Feet, Chains, and Yards 337 

Tables of Distances in Yards, Chains, and Feet. . « 237 

GRADIENTS. 

Tables of Gradients 239 

Explanation of the Table of Gradients , 246 

Form for arranging lists of Gradients 247 

Rule to find the Gravitation of 1 Ton on a given Incline 247 

Form for arranging the Power required to overcome Gravitation. . 249 
Rule for computing the average Gravitation of 1 Ton over a series 
of Inclines 250 

POWER. 

Essay on the Power required to move a given Load 
ALONG Inclined Planes , 251 



▼lU CONTBNTS. 

Page. 

Reristance due to Friction 251 

Resistance of the Air 252 

Table for computing the Resistance of the Air 260 

Resistance due to the Gradient 261 

General Rule for the total Resistance 262 

Power of the Engines required to draw a given Load 263 

Adhesion of the Engine Wheels to the Rails 264 

Vaporization of Water of the Engines 265 

Power of the Engines 266 

Power on an Incline compared with that on a Level 267 

The duty one Engine can perform 267 

Table of Vaporization of Water^ and the Power of the Enjj^e under 

various Pressures 268 

Capability of an Engine (Causes of variation of) , 268 

Accidental Causes of Resistance 270 

Resistance of the Wind 270 

Rules for calculating the Resistance of the Wind 274 

Resistance of Curves 275 

Centrifugal Force round Curves 276 

Elevation of Outer Rail round Curves . . 277 

Impingence of the Wheels against the Rails 278 

Conical Form of the Tires of the Wheels 279 

Rule for finding the Force of Impingence against the Rails 281 

Rule to find the number of Oscillations per second due to the 

Impingence 281 

Rule to find the Resistance of a Curve (generally) 283 

Table for estimating the Variations of the above Rules/ consequent 

upon an increased distance between the front and hind wheels 285 

Rule to find the Resistance of a Straight line 285 

Conclusion 28 5 



Errata : 

P^^ 237— /or << yards into chains and feet/' read ''feet into chains 

and yards." 
Page 238— /or ''feet into chains and yards/' read "yards into 

chains and feet/' 



ERRATA. 

In page v., after \\ to, insert 1. 

187> /tne ;j, after W and B^, instead qf + read x . 

193, Une 24, instead ofr + 9, read r ± s. 

198, Une 13, instead of r •»■ '1, reoc^ i^ + 1* 

200, ftite 20, instead of L, reocf Z. 

202, tt»e 3, instead of r^ + 1, reod rs \/r«~^ 1. 

202, rule 12, in the margin, instead of s, reoc^ 3. 

208, Une 13, ti»feaii of chasm or, read chance of. 

210, Une 11, instead of WBS, read were. 

231, role 3, in the margin, instead of AD'' AB^ 
rearf AD« — AB». 

236, Une 21, instead of 2|, read 2f . 

260, Itne 27, instead qf role, reacf rise. 



ft 

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63 


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56 


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56 


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27 


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15.5740 


29 


22.4212 


59 


70.1990 


29 


3.8935 


69 


16.1167 


30 


23.6111 


60 


72.2222 


30 


4.1667 


60 


16.6666 





(iv. 


) 




BASE 20 SLOPE 1 to 1. 




1. 


Add. 


■4^ 

-a 

•53 


Add. 




Deduct. 


<4H 


Deduct. 


1 




W 




Q* 




Q 




.3888 


31 


29.2777 


1 


.0062 


31 


5.9321 


2 


.8148 


32 


30.8148 


2 


.0247 


32 


6.3210 


3 


1.2777 


33 


32.3888 


3 


.0555 


33 


6.7222 


4 


1.7777 


34 


34.0000 


4 


.0988 


34 


7.1358 


5 


2.3148 


35 


35.6481 


5 


.1543 


35 


7.5617 


6 


2.8888 


36 


37.3333 


6 


.2222 


36 


8.0000 


7 


3.5000 


37 


39.0555 


7 


.3025 


37 


8.4506 


8 


4.1481 


38 


40.8148 


8 


.3951 


38 


8.9135 


9 


4.8333 


39 


42.6111 


9 


.5000 


39 


9.3888 


10 


6.6666 


40 


44.4444 


10 


.6173 


40 


9.8765 


11 


6.3148 


41 


46.3148 


11 


.7469 


41 


10.3765 


12 


7.1111 


42 


48.2222 


12 


.8889 


42 


10.8888 


13 


7.9444 


43 


50.1666 


13 


1.0432 


43 


11.4135 


14 


8.8148 


44 


52.1481 


14 


1.2099 


44 


11.9506 


15 


9.7222 


46 


54.1666 


16 


1.3889 


45 


12.5000 


16 


10.6666 


46 


66.2222 


16 


1.5802 


46 


13.0617 


17 


11.6481 


47 


58.3148 


17 


1.7839 


47 


13.6358 


18 


12.6666 


48 


60.4444 


18 


2.0000 


48 


14.2222 


19 


13.7222 


49 


62.6111 


19 


2.2284 


49 


14.8209 


20 


14.8148 


50 


64.8148 


20 


2.4691 


50 


15.4321 


21 


15.9444 


51 


67.0555 


21 


2.7222 


51 


16.0555 


22 


17.1111 


52 


69.3333 


22 


2.9876 


52 


16.6913 


23 


18.3148 


53 


71.6481 


23 


3.2654 


53 


17.3395 


24 


19.5555 


54 


74.0000 


24 


3.5555 


54 


18.0000 


25 


20.8333 


66 


76.3888 


25 


3.8530 


66 


18.6728 


26 


22.1481 


66 


78.8148 


26 


4.1728 


6Q 


19.3580 


27 


23.5000 


61 


81.2777 


27 


4.5000 


61 


20.0555 


28 


24.8888 


58 


83.7777 


28 


4.8395 


58 


20.7654 


29 


26.3148 


59 


86.3148 


29 


5.1913 


59 


21.4876 


30 


27.7777 


60 


88.8888 


30 5.5555 

• 


60 


22.2-222 



I 




(V. 


) 


BASE 20— SLOPE li to 1. 




Add. 


4d 

t 


.Add. 


o • 


Deduct. 


<4H 

"a < 

*a S Dedact. 


B 




31 




o* 




Q°' 


1 


.3935 


33.7268 


1 


.0077 


31 


7.4151 


2 


.8333 


32 


36.6566 


2 


.0309 


32 


7.9012 


3 


1.3194 


33 


37.4305 


3 


.0694 


33 


8.4027 


4 


1.8518 


34 


39.3518 


4 


.1234 


34 


8.9197 


5 


2.4305 


35 


41.3194 


5 


.1928 


35 


9.4621 


6 


3.0555 


36 


43.3333 


6 


.2778 


36 


10.0000 


7 


3.7268 


37 


45.3935 


7 


.3781 


37 


10.5632 


8 


4.4444 


38 


47.5000 


8 


.4938 


38 


11.1419 


9 


5.2083 


39 


49.6527 


9 


.6250 


39 


11.7361 


10 


6.0185 


40 


51.8518 


10 


.7716 


40 


12.3456 


11 


6.8750 


41 


54.0972 


11 


.9336 


41 


12.9706 


12 


7.7777 


42 


56.3888 


12 


1.1111 


42 


13.6111 


13 


8.7268 


43 


58.7268 


13 


1.3040 


43 


14.2680 


14 


9.7222 


44 


61.1111 


14 


1.5123 


44 


14.9382 


15 


10.7638 


45 


63.5416 


15 


1.7361 


45 


15.6250 


16 


11.8518 


46 


66.0185 


16 


1.9753 


46 


16.3271 


17 


12.9861 


47 


68.5416 


17 


2.2299 


47 


17.0447 


18 


14.1666 


48 


71.1111 


18 


2.5000 


48 


17.7777 


19 


15.3935 


49 


73.7268 


19 


2.7855 


49 


18.5262 


20 


16.6666 


50 


76.3888 


20 


3.0864 


60 


19.2901 


21 


17.9861 


51 


79.0972 


21 


3.4028 


51 


20.0694 


22 


19.3518 


52 


81.8518 


22 


3.7346 


52 


20.8641 


23 


20.7638 


53 


84.6527 


23 


4.0818 


53 


21.6743 


24 


22.2222 


54 


87.5000 


24 


4.4444 


54 


22.5000 


25 


23.7268 


55 


90.3935- 


25 


4.8228 


55 


23.3410 


26 


25.2777 


56 


93.3333 


26 


5.2160 


56 


24.1975 


27 


26.8750 


57 


96.3194 


27 


5.6250 


57 


25.<694 


28 


28.5185 


58 


99.3518 


28 


6.0494 


68 ;25.9567 


29 


30.2083 


59 


102.4305 


29 


6.4891 59 


26.8595 


30 

1 - 


31.9444 


60 


105.5555 


30 


6.9444 60 


27.7777 





(iv.) 


1 




BASE 20 SLOPE 1 to 1. 






1 


Add. 


-a 


Add. 


4m 


Deduct. 


<M 


Deduct. 


1 


X 

1 




K 




0* 

1 




p 




1 


.3888 


31 


29.2777 


.0062 


31 


5.9321 


1 


2 


.8148 


32 


30.8148 


2 


.«J247 


32 


6.3210 




3 


1.2777 


33 


32.3888 


3 


.0555 


33 


6.7222 




4 


1.7777 


34 


34.0000 


4 


.0988 


34 


7.1358 




5 


2.3148 


35 


35.6481 


5 


.1543 


36 


7.5617 




6 


2.8888 


36 


37.3333 


6 


.2222 


36 


8.0000 




7 


3.5000 


37 


39.0555 


7 


.3026 


37 


8.4506 




8 


4.1481 


38 


40.8148 


8 


.3951 


38 


8.9135 




9 


4.8333 


39 


42.6111 


9 


.5000 


39 


9.3888 


1 


10 


5.6665 


40 


UMU 


10 


.6173 


40 


9.8765 




11 


6.3148 


41 


46.3148 


11 


.7469 


41 


10.3765 




12 


7.1111 


42 


48.2222 


12 


.8889 


42 


10.8888 




13 


7.9444 


43 


50.1666 


13 


1.0432 


43 


11.4135 




14 


8.8148 


44 


52.1481 


14 


1.2099 


44 


11.9506 




15 


9.7222 


45 


54.1666 


15 


1.3889 


45 


12.5000 




16 


10.6666 


46 


66.2222 


16 


1.5802 


46 


13.0617 




17 


11.6481 


47 


58.3148 


17 


1.7839 


47 


13.6358 




18 


12.6666 


48 


60.4444 


18 


2.0000 


48 


14.2222 




19 


13.7222 


49 


62.6111 


19 


2.2284 


49 


14.8209 




2U 


14.8148 


50 


64.8148 


20 


2.4691 


60 


15.4321 




21 


15.9444 


51 


67.0555 


21 


2.7222 


51 


16.0555 




22 


17.1111 


52 


69.3333 


22 


2.9876 


62 


16.6913 


' 


23 


18.3148 


53 


71.6481 


23 


3.2654 


63 


17.3395 




24 


19.5555 


54 


74.0000 


24 


3.5555 


54 


18.0000 




25 20.8333 


55 


76.3888 


23 


3.8530 


55 


18.6728 




26 22.1481 


56 


78.8148 


26 


4.1728 


56 


19.3580 




27 '23.6000 


57 


81.2777 


27 


4.5000 


57 


20.0555 




28 24.8888 


58 


83.7777 


28 


4.8395 


68 


20.7054 




29 26.3148 


59 


86.3148 


29 


5.1913 


69 


21.1876 




30 !27.7777 60 


88.8888 


30 


5.5555 

* 


60 


22.2i22 













BASE 20— SLOPE l^ to 1. 



** 

t 

93 


Add. 


** 

1 
31 


1 


.3935 


2 


.8333 


32 


3 


1.3194 


33 


4 


1.8518 


34 


5 


2.4305 


35 


6 


3.0555 


36 


7 


3.7268 


37 


8 


4.4444 


38 


9 


5.2083 


39 


10 


6.0185 


40 


11 


6.8750 


41 


12 


7.7777 


42 


13 


8.7268 


43 


14 


9.7222 


44 


15 


10.7638 


45 


16 


11.8518 


46 


17 


12.9861 


47 


18 


14.1666 


48 


19 


15.3935 


49 


20 


16.0GG6 


50 


21 


17.9861 


61 


22 


19.3518 


52 


23 


20.7638 


53 


24 


22.2222 


54 


25 


23.7268 


55 


26 


25.2777 


56 


27 


26.8750 


57 


28 


28.5185 


58 


29 


30.2083 


59 


30 


^1.9444 


60 



Add. 



33.7268 
35.5555 
37.4305 
39.3518 
41.3194 

43.3333 

45.3935 
47.5000 
49.6527 
51.8518 

54.0972 
56.3888 
58.7268 
61.1111 
63.5416 

66.0185 
68.5416 
71.1111 
73.7268 

76.3888 

79.0972 
81.8518 
84.6527 
87.5000 
90.3935 

93.3333 

96.3194 

99.3518 

102.4305 

105.5555 



.^5 



1 

2 
3 
4 
5 

6 
7 
8 
9 
10 

11 
12 
13 
14 
15 

16 
17 
18 
19 
20 

21 
22 
23 
24 
25 

26 

27 
28 
29 
30 



Dfldaot. 



.0077 
.0309 
.0694 
.1234 
.1928 

.2778 
.3781 
.4938 
.6250 
.7716 

.9336 
1.1111 
1.3040 
1.5123 
1.7361 

1.9753 
2.2299 
2.5000 
2.7855 
3.0864 

3.4028 
3.7346 
4.0818 
4.4444 

4.8228 

5.2160 
5.6250 
6.0494 
6.4891 
6.9444 






Dedact. 



31 
32 
33 
34 
35 

86 
37 
38 
39 

40 

41 
42 
43 
44 
45 



7.4151 
7.9012 
8.4027 
8.9197 
9.4521 

10.0000 
10.5632 
11.1419 
11.7361 
12.3456 

12.9706 
13.6111 
14.2680 
14.9382 
15.6250 



46 16.3271 

47 17.0447 

48 '17.7777 

49 118.5262 

50 19.2901 

51 20.0694 

52 20.8641 

53 21.6743 

54 ,22.5000 
5o ,23.3410 

56 124.1975 

57 i25.< 694 

58 |25.9567 

59 126.8595 

60 '27.7777 





(vi. 


) 




BASE 20— SLOPE IJ to 1. 




-a 


Add. 


1 

V 


Add. 




Deduct. 




Deduct. 


1 




» 




1 




Q* 




.3981 


31 


38.1759 


.0092 


31 


8.8981 


2 


.8518 


32 


40.2962 


2 


.0370 


32 


9.^815 


3 


1.3611 


33 


43.4722 


3 


.0833 


33 


10.0833 


4 


1.9259 


34 


44.7037 


4 


.1480 


34 


10.7037 


5 


2.5462 


35 


46.9907 


5 


.2313 


35 


11.3425 


6 


3.2222 


36 


49.3333 


6 


.3333 


36 


12,0000 


7 


3.9537 


37 


51.7314 


7 


.4537 


37 


12.6759 


8 


4.7407 


38 


54.1851 


8 


.5926 


38 


13.3703 


9 


5.5833 


39 


56.6944 


9 


.7500 


39 


14.0833 


10 


6.4814 


40 


59.2592 


10 


.9259 


40 


14.8148 


11 


7.4351 


41 


61.2129 


11 


1.1203 


41 


15.5648 


12 


8.4444 


42 


64.5555 


12 


1.3333 


42 


16.3333 


13 


9.5092 


43 


67.2870 


13 


1.5648 


43 


17.1202 


14 


10.6296 


44 


70.0740 


14 


1.8148 


44 


17.9259 


15 


11.8055 


45 


72.9166 


15 


2.0833 


45 


18.7500 


16 


13.0370 


46 


75.8148 


16 


2.3704 


46 


19.5925 


17 


14.3240 


47 


78.7685 


17 


2.6759 


47 


20.4537 


18 


15.6666 


48 


81.7777 


18 


3.0000 


48 


21.3333 


19 


17.0648 


49 


84.8425 


19 


3.3426 


49 


22.2314 


20 


18.5185 


50 


87.9629 


20 


3.7037 


50 


23.1481 


21 


20.0277 


51 


91.1388 


21 


4.0833 


51 


24.0833 


22 


21.5925 


62 


94..3703 


22 


4.4815 


52 


25.0370 


23 


23.2129 


53 


97.6574 


23 


4.8981 


53 


26.0092 


24 


24.8888 


•54 


101.0000 


24 


5.3333 


54 


27.0000 


25 


26.6203 


55 


104.3981 


25 


5.7869 


55 


28.0092 


26 


28.4074 


56 


107.8518 


26 


6.2592 


56 


29.0370 


27 


30.2500 


57 


111.3611 


27 


6.7500 


51 


30.0833 


28 


32.1481 


58 


114.92.59 


28 


7.2592 


58 


31.1481 


29 


34.1018 


59 


118.5462 


29 


7.7869 


59 


32.2314 


30 


36.1111 


60 


122.2222 


30 


8.3333 60 


33.3333 



J 





( 'ii. ) 


BASE 20— SLOPE IJ to 1. 


4J 

1 


Add. 


•s 


Add. 


S3 33 


• 

Deduct. 




Deduct. 


1 




X 

31 




Q* 




o* 


• 


.4027 


42.6250 


1 


.0108 


31 


10.3811 


2 


.8703 


32 


43.0370 


2 


.0432 


32 


11.0609 


3 


1.4027 


33 


47.5138 


3 


.0972 


33 


11.7638 


4 


2.0000 


34 


50.0555 


4 


.1728 


34 


12.4876 


5 


2.6620 


35 


52.6620 


6 


.2700 


36 


13.2330 


6 


3.3888 


86 


65.3333 


6 


.3889 


36 


14.0000 


7 


4.1805 


37 


68.0694 


7 


.5293 


37 


14.7885 


8 


6.0370 


38 


60.8703 


8 


.6913 


38 


15.5987 


9 


6.9683 


39 


63.7361 


9 


.8750 


39 


16.4305 


10 


6.9444 


40 


66.6666 


10 


1.0802 


40 


17.2839 


11 


7.9953 


41 


69.6620 


11 


1.3071 


41 


18.1589 


12 


9.1111 


42 


72.7222 


12 


1.5555 


42 


19.0555 


13 


10.2916 


43 


75.8472 


13 


1.8256 


43 


19.9747 


14 


11.6370 


44 


79.0370 


14 


2.1173 


44 


20.9135 


16 


12.8472 


45 


82.2916 


15 


2.4305 


45 


21 .8750 


16 


14.2222 


46 


86.6111 


16 


2.7654 


46 


22.8580 


17 


16.6620 


47 


8S.9953 


17 


3.1219 


47 


23.8626 


18 


17.1666 


48 


92.4444 


18 


3.5000 


48 


24.8888 


19 


18.7361 


49 


95.9583 


19 


3.8997 


49 


25.9367 


20 


20.3703 


60 


99.6.!f70 


20 


4.3210 


50 


27.0061 


21 


22.0694 


61 


103.1805 


21 


4.7639 


51 


28.0972 


22 


23.8333 


52 


106.8888 


22 


5.2284 


62 


29.2098 


23 


26.6620 


53 


110.6620 


23 


5.7145 


53 


30.3441 


24 


27.6665 


54 


114.5U00 


24 


6.2222 


64 


31.5000 


25 


29.5138 


65 


118.4027 


25 


6.7516 


56 


32.6774 


26 


31.6370 


56 


122.3703 


26 


7.3024 


66 


33.8765 


27 


33.6250 


67 


126.4027 


27 


7.8750 


57 


35.0972 


28 


35.7777 


68 


130.6000 


28 


8.4691 


58 


36.3394 


29 


37.9932 


69 


134.6620 


29 


9.0848 


59 


37.6033 


30 


40.27771 60 


138.8888 


SO 


9.7222 


60 38.8888 











III. ) 










(vi 




BASE 20— SLOPE 2 to 1. 


1 


Add. 


1 

•s 

K 

31 


Add. 


Dif. of 
-- Hts. 


Dedaet. 


Dif.of 
Hts. 


Deduct. 


.4074 


47.0740 


.0123 


31 


11.8642 


2 


.8888 


32 


49.7777 


2 


.0494 


32 


12.6419 


3 


1.4444 


33 


62.5555 \ 


3 


.1111 


33 


13.4444 


4 


2.0740 


34 


55.4074 


4 


.1975 


34 


14.2716 


5 


2.7777 


35 


58.3333 


5 


.3086 


35 


15.1234 


6 


3.6555 


36 


61.3333 


6 


.4444 


36 


16.0000 


7 


4.4074 


37 


64.4074 


7 


.6049 


37 


16.9012 


8 


5.3333 


38 


67.5555 


8 


.7901 


38 


17.8271 


9 


6.3333 


39 


70.7777 


9 


1.0000 


39 


18.7777 


10 


7.4074 


40 


74.0740 


10 


1.2346 


40 


19.7530 


11 


8,5555 


41 


77.4444 


11 


1.4938 


41 


20.7530 


12 


9.7777 


42 


80.8888 


12 


1.7778 


42 


21.7777 


13 


11.0740 


43 


84.4074 


13 


2.0864 


43 


22.8271 


14 


12.4444 


44 


88.0000 


14 


2.4197 


44 


23.9012 


15 


13.8888 


45 


91.6666 


16 


2.7778 


45 


25.0000 


16 


15.4074 i 46 


95.4074 


16 


3.1605 


46 


26.1234 


17 


17.0000 


47 


99.2222 


17 


3.5679 


47 


27.2716 


18 


18.6666 


48 


103.1111 


18 


4.0000 


48 


28.4444 


19 


20.4074 


49 


107.0740 


19 


4.4568 


49 


29.6420 


20 


22.2222 


50 


111.1111 


20 


4.9382 


50 


30.8642 


21 


24.1111 


51 


115.2222 


21 


5.4444 


51 


32.1111 


22 


26.0740 


52 


119.4074 


22 


5.9753 


62 


33.3827 


23 


28.1111 


53 


123.6666 


23 


6.5309 


53 


34.6790 


24 


30.2222 


54 


128.0000 


24 


7.1111 


54 


36.0000 


26 


32.4074 


55 


132.4074 


25 


7.7160 


55 


37.3456 


26 


34.6666 


56 


136.8888 


26 


8.3457 


66 


38.7160 


27 


37.0000 


57 


141.4444 


27 


9.0000 


67 


40.1111 


28 


39.4074 58 


146.0740 


28 


9.6790 


68 


41.5308 


29 


41.8888 , 59 


150.7777 


29 


10.3827 


69 


42.9753 


30 44.4444160 


155.6555 


130 


11.1111 60 144.4444 


,. .. , 



.J 



(«.) 


BASE 20-SLOPE ^ to 1. 


t 

SB 


Add. 


1 


Add. 


3^ 


Dedvct 


^a 
§« 


Deduct. 


T 


.4166 


31 


66.9722 


1 


.0154 


31 


148302 


2 


.9259 


32 


59.2592 


2 


.0617 


32 


15.8016 


3 


1.5277 


33 


62.6388 


3 


.1389 


33 


m8066 


4 


2.2222 


34 


66.1111 


4 


.2468 


34 


17.8395 


5 


a0092 


36 


69.6769 


6 


.3857 


35 


18.9043 


6 


3.8888 


36 


73.3333 


6 


.6656 


36 


20.0000 


7 


4.8611 


37 


77.0833 


7 


.7561 


37 


21.1265 


8 


5.9259 


38 


80.9269 


8 


.9876 


38 


22.2839 


9 


7.0833 


39 


84.8611 


9 


1.2600 


39 


23.4722 


10 


8.3333 


40 


88.8888 


10 


1.5432 


40 


24.6913 


11 


9.6759 


41 


93.0092 


11 


1.8673 


41 


25.9413 


12 


11.1111 


42 


97.2222 


12 


2.2222 


42 


27.2222 


13 


12.6388 


43 


101.5277 


13 


2.6080 


43 


28.5360 


14 


14.2592 


44 


105.9269 


14 


3.0247 


44 


29.8765 


16 


16.9722 


45 


110.4166 


15 


3.4722 


46 


31.2600 


16 


17.7777 


46 


116.0000 


16 


3.9506 


46 


32.6643 


17 


19.6759 


47 


119.6759 


17 


4.4599 


47 


34.0895 


18 


21.6666 


48 


124.4444 


18 


6.0000 


48 


35.5555 


19 


23.7500 


49 


129.3055 


19 


5.5710 


49 


37.0524 


20 


25.9259 


50 


134.2692 


20 


6.1728 


50 


38.5802 


21 


28.1944 


51 


139.3055 


21 


6.8055 


61 


40.1388 


22 


30.5555 


52 


144.4444 


22 


7.4691 


62 


41.7283 


23 


33.0092 


53 


149.6759 


23 


8.1636 


53 


43.3487 


24 


35.5555 


54 


155.0000 


24 


8.8889 


54 


45.0000 


25 


38.1944 


55 


160.4166 


25 


9.6466 


66 


46.6820 


26 


40.9259 


56 


165.9269 


26 


10.4321 


66 


48.3950 


27 


43.7500 


57 


171.5277 


27 


11.2500 


57 


50.1388 


28 


46.6666 


58 


177.2222 


28 


12.0687 


68 


51.9136 


29 


49.6759 


59 


183.0092 


29 


12.9782 


59 


53.7191 


30 


52.7777 


60 188.8888 


30 


13.8889 


60 
c 


55.5556 



( X. ) 


BASE 20— SLOPE 3 to 1, 


•s 


Add. 


J 


Add. 


l^ 

^m 


Deduct. 


**9 

S5B3 


1 

Deduct. 


as 




s 


1 


Q 




a 




1 


.4259 


31 


64.8703 


1 


.0186 


31 


17.7963 


2 


.9629 


32 


68.7407 


2 


.0740 


32 


18.9630 


3 


1.6111 


33 


72.7222 


3 


.1667 


33 


20.1666 


4 


2.3703 


34 


76.8148 


4 


.2961 


34 


21.4074 


3 


3.2407 


35 


81.0186 


5 


.4628 


35 


22.6851 


6 


4.2222 


36 


86.3333 


6 


.6667 


36 


24.0000 


7 


5.3148 


37 


89.7592 


7 


.9074 


37 


25.3618 


8 


6.5185 


38 


94.2962 


8 


1.1852 


38 


26.7407 


9 


7.8333 


39 


98.9444 


9 


1.5000 


39 


28.1666 


10 


9.2692 


40 


103.7037 


10 


1.8518 


40 


29.6296 


11 


10.7962 


41 


108.6740 


11 


2.2407 


41 


31.i296 


12 


12.4444 


42 


113.5655 


12 


2.6667 


42 


32.6666 


13 


14.2037 


43 


118.6481 


13 


3.1296 


43 


34.2405 


14 


16.0740 


44 


123.8618 


14 


3.6296 


44 


36.8518 


15 


18.0566 


45 


129.1666 


16 


4.1667 


45 


37.6000 


16 


20.1481 


46 


134.5926 


16 


4.7407 


46 


39.1851 


17 


22.3518 


47 


140.1296 


17 


5.3518 


47 


40.9074 


18 


24.6666 


48 


145.7777 


18 


6.0000 


48 


42.6666 


19 


27.0925 


49 


151.6370 


19 


6.6852 


49 


44.4629 


20 


29.6296 


60 


167.4074 


20 


7.4074 


50 


46.2963 


21 


32.2777 


61 


163.3888 


21 


8.1667 


61 


49.1666 


22 


35.0370 


62 


169.4814 


22 


8.9629 


62 


50.0740 


23 


37.9074 


53 


175.6851 


23 


9.7962 


53 


62.0186 


24 


40.8888 


54 


182.0000 


24 


10.6667 


54 


54.0000 


25 


-^.9814 


65 


188.4259 


25 

f * 


11.6741 


65 


56.0184 


26 


47.1851 


66 


194.9629 


26 


12.5184 


56 '58.0640 


27 


60.6000 


57 


201.6111 


27 


13.5000 


57 60.1666 


28 


53.9259 


58 


206.3703 


28 


14.6185 


58 62.2962 


29 


57.4629 


69 


215.2407 


29 


15.5739 


59 64.4629 


30 


61.1111 


60 


222.2222 


30 


16.6667 


60 ;66.6666 



J 









(xi.) 






BASE 21— SLOPS 


1 1 to 1. 


t 

X 

1 


Add. 


4J 

t 
X 


Add. 


Dif.of 
Hts. 


Dednct. 


Dif.of 
Hts. 


Deduct. 


.3935 


31 


16.5046 


1 


.0016 


31 


1.4830 


2 


.7962 


32 


17.1861 


2 


.00®J 


32 


1.5802 


3 


1.2083 


33 


17.8750 


3 


.0139 


33 


1.6805 


4 


1.6296 


34 


18.5740 


4 


.0246 


34 


1.7839 


5 


2.0601 


35 


19.2824 


6 


.0385 


35 


1.8904 


6 


2.5000 


36 


20.0000 


6 


.0555 


36 


2.0000 


7 


2.9490 


37 


20.7268 


7 


.0756 


37 


2.1126 


8 


3.4074 


38 


21.4629 


8 


.0988 


38 


2.2284 


9 


3.8750 


39 


22.2083 


9 


.1250 


39 


2.3472 


10 


4.3518 


40 


22.9629 


10 


.1543 


40 


2.4691 


11 


48379 


41 


23.7268 


11 


.1867 


41 


2.6941 


12 


5.3333 


42 


24.5000 


12 


.2222 


42 


2.7222 


13 


5.8379 


43 


26.2824 


13 


.2608 


43 


2.8546 


14 


6.3518 


44 


26.0740 


14 


.3025 


44 


2.9876 


16 


6.8750 


46 


26.8750 


15 


.3472 


46 


3.1250 


16 


7.4074 


46 


27.6851 


16 


.3951 


46 


3.2654 


17 


7.9490 


47 


28.6046 


17 


.4460 


47 


3.4068 


18 


8.5000 


48 


29.3333 


18 


.5000 


48 


3.6556 


19 


9.0601 


49 


30.1712 


19 


.6571 


49 


3.7062 


20 


9.6296 


60 


31.0186 


20 


.6173 


50 


3.8580 


21 


ia2083 


51 


31.8760 


21 


.6805 


51 


4.0139 


22 


10.7962 


52 


32.7407 


22 


.7469 


52 


4.1728 


23 


11.3935 


53 


33.6157 


23 


.8163 


63 


4.33^ 


24 


12.0000 


54 


34.6000 


24 


.8889 


64 


4.5000 


25 


12.6157 


65 


35.3935 


25 


.9647 


55 


4.6682 


26 


13.2407 


56 


36.2962 


26 


1.0432 


66 


4.8396 


27 


13.8750 


57 


37.2083 


27 


1.1250 


67 


5.0139 


28 


14.5186 


58 


38.1296 


28 


1.2099 


58 


6.1913 


29 


15.1712 


69 


39.0601 


29 


1.2978 


59 


6.3719 


30 


15.8333 


60 


40.0000 


30 


1.3889 


60 


5,5665 





(xii.) 




BASE 21— SLOPE 


i to 1. 


} 


Add. 


} 


Add. 


if. of 
Hts. 


Dednct. 


■5 Eu 


Deduct. 


1 




S 




Q 




Q 




.3981 


31 


20.9537 


1 


.0031 


31 


2.9660 


2 


.8148 


32 


21.9259 


2 


.0123 


32 


3.1605 


3 


1.2500 


33 


22.9166 


3 


.0278 


33 


3.6111 


4 


1.7037 


34 


23.9259 


4 


.0494 


34 


3.5679 


5 


2.1759 


35 


24.9537 


5 


.0772 


36 


3.7808 


6 


2.6666 


36 


26.0000 


6 


.1111 


36 


4.0000 


7 


3.1759 


37 


27.0648 


7 


.1512 


37 


42263 


8 


3.7037 


38 


28.1481 


8 


.1975 


38 


44568 


9 


4.2500 


39 


29.2500 


9 


.2500 


39 


46944 


10 


4.8148 


40 


30.3703 


10 


.3086 


40 


49383 


11 


5.3981 


41 


31.5092 


11 


.3734 


41 


5.1883 


12 


6.0000 


42 


32.6666 


12 


.4444 


42 


6.4444 


13 


6.6203 


43 


33.8425 


13 


.5216 


43 


6.7076 


14 


7.2592 


44 


35.0370 


14 


.6049 


44 


5.9753 


15 


7.9166 


45 


36.2600 


15 


.6944 


46 


6.2500 


16 


8.5925 


46 


37.4814 


16 


.7901 


46 


6.5308 


17 


9.2870 


47 


38.7314 


17 


.8920 


47 


6.8179 


18 


10.0000 


48 


40.0000 


18 


1.0000 


48 


7.1111 


19 


10.7314 


49 


41.2870 


19 


1.1142 


49 


7.4104 


20 


11.4814 


60 


42.5925 


20 


1.2346 


50 


7.7160 


21 


12.2500 


61 


43.9166 


21 


1.3611 


61 


8.0277 


22 


13.0370 


52 


46.2592 


22 


1.4938 


52 


8.3456 


23 


13.8425 


53 


46.6203 


23 


1.6327 


53 


8.6697 


24 


14.6666 


54 


48.0000 


24 


1.7778 


64 


9.0000 


25 


15.5092 


55 


49.3982 


25 


1.9295 


55 


9.3364 


26 


16.3703 


66 


60.8148 


26 


2.0864 


56 


9.6790 


27 


17.2500 


57 


52.2500 


27 


2.2500 


67 


10.0277 


28 


18.1481 


58 


53.7037 


28 


2.4197 


58 


10.3827 


29 


19.0648 


59 


55.1769 


29 


2.5956 


59 


10.7438 


30 




20.0000 


60 


56.6666 


30 


2.7778 


60 11.1111 



( xiii. ) 


BASE 21 SLOPE f to 1. 


} 


Add. 


•8 


Add. 


if. of 


1 

Deduct. 




Dednct. 


« 




B 




Q 




31 




1 


.4027 


31 


26.4027 


1 


.0046 


4.4490 


2 


.8333 


32 


26.6666 


2 


.0186 


32 


4.7407 


3 


1.2916 


33 


27.9583 


3 


.0416 


33 5.0416 


4 


1.7777 


34 


29.2777 


4 


.0740 


34 5.3518 


5 


2.2916 


35 


30.6250 


6 


.1157 


36 


6.6712 


6 


2.8333 


36 


32.0000 


6 


.1667 


36 


aoooo 


7 


3.4027 


37 


33.4027 


7 


.2268 


37 


&3379 


8 


4.0000 


38 


34.8333 


8 


.2963 


38 


&6861 


9 


4.6250 


39 


36.2916 


9 


.3750 


39 


7.0416 


10 


5.3333 


40 


37.7777 


10 .4630 


40 


7.4074 


11 


5.9583 


41 


39.2916 


11 .5602 


41 


7.7824 


12 


6.6666 


42 


40.8333 


12 


.6667 


42 


8.1666 


13 


7.4027 


43. 


^.4027 


13 


.7824 


43 


8.6612 


14 


8.1666 


44 


44.0000 


14 


.9074 


44 


8.9629 


16 


8.9583 


45 


45.6260 


15 


1.0417 

• 


45 


9.3760 


16 


9.7777 


46 


47.2777 


16 


1.1852 


46 


9.7962 


17 


10.6250 


47 


48.9583 


17 


1.3379 


47 


10.2268 


18 


11.5000 


48 


50.6666 


18 


1.6000 


48 


10.6666 


19 


12.4027 


49 


52.4027 


19 


1.6713 


49 


11.1167 


20 


13.3333 


50 


64.1666 


20 


1.8618 


50 


11.6740 


21 


14.2916 


51 


55.9583 


21 


2.0417' 


51 


12.0416 


22 


15.2777 


52 


57.7777 


22 


2.2407 


52 


12.6186 


23 


16.2916 


53 


69.6250 


23 


2.4491 


63 


13.0046 


24 


17.3333 


54 


61.5000 


24 


2.6667 


54 


13.5000 


25 


18.4027 


55 63.4027 


25 


2.8935 


65 


14.0046 


26 


19.5000 


56 


65.3333 


26 


3.1296 


56 


146186 


27 


20.6250 


57 


67.2916 


27 


3.3760 


67 


16.0416 


28 


21.7777 


58 


69.2777 


28 


3.6296 


68 


16.6740 


29 


22.9583 


59 


71.2916 


29 


3.8936 


69 


16.1157 


30 


24.1666 


60 


73.3333 


30 


4.1667 


60 


116.6666 





( xiv. ) 


• 


BASE 21— SLOPE 1 to 1. 


1 

V 


Add. 


■s 


Add. 




Deduct. 


S3 as 


Deduct. 


1 




a 




Q 




Q 




.4074 


31 


29.8518 


1 


.0062 


31 


6.9321 


2 


.8518 


32 


31.4074 


2 


.0247 


32 


6.3210 


3 


1.3333 


33 


33.0000 


3 


.0555 


33 


6.7222 


4 


1.8518 


34 


34.6296 


4 


.0988 


34 


7.1358 


5 


2.4074 


35 


36.2962 


5 


.1543 


35 


7.6617 


6 


3.0000 


36 


38.0000 


6 


.2222 


36 


8.0000 


7 


3.6296 


37 


39.7407 


7 


.3025 


37 


8.4506 


8 


4.2962 


38 


41.6185 


8 


.3951 


38 


8.9135 


9 


5.0000 


39 


43.3333 


9 


.5000 


39 


9.3888 


10 


6.7407 


40 


45.1861 


10 


.6173 


40 


9.8765 


11 


6.5185 


41 


47.0740 


11 


.7469 


41 


10.3766 


12 


7.3333 


42 


49.0000 


12 


.8889 


42 


10.8888 


13 


8.1851 


43 


60.9629 


13 


1.0432 


43 


11.4135 


14 


9.0740 


44 


52.9629 


14 


1.2099 


44 


11.9606 


16 


10.0000 


45 


56.0000 


15 


1.3889 


45 


12.5000 


16 


10.9629 


46 


67.0740 


16 


1.5802 


46 


13.0617 


17 


11.9629 


47 


59.1851 


17 


1.7839 


47 


13.6358 


18 


13.0000 


48 


61.3333 


18 


2.0000 


48 


14.2222 


19 


14.0740 


49 


63.6185 


19 


2.2284 


49 


14.8209 


20 


15.1851 


50 


65.7407 


20 


2.4691 


60 


15.4321 


21 


ia3333 


51 


68.0000 


21 


2.7222 


61 


16.a'565 


22 


17.6186 


62 


70.2962 


22 


2.9876 


52 


16.6913 


23 


18.7407 


53 


72.6296 


23 


3.2654 


53 


17.3396 


24 


20.0000 


64 


76.0000 


24 


3,5566 


64 


18.0000 


25 


21.2962 


55 


77.4074 


25 


3.8530 


55 


18.6728 


26 


22.6296 


56 


79.8518 


26 


4.1728 


66 


19.3580 


27 


24.0000 


57 


82.3333 


27 


4.5000 


57 


20.0656 


28 


26.4074 


58 


85.8618 


28 


4.8395 


58 


20.7654 


29 


26.8518 69 


87.4074 


29 


5.1913 


69 


21.4876 


30 


28.3333 60 


90.0000 


30 


5.5666 60 


22.2222 







(XV 


•) 


1 

■ 1 




BASE 21 SLOPE 


li to 1. 


t 


Add. 


1 

X 


Add. 


° a 

ran 

a 

1 


Deduct. 


31 


1 

Deduct. 

1 


1 


.4120 


31 


34.3009 


.0077 


7.4161 


2 


.8703 


32 


36.1481 


2 


.0309 


32 


7.9012 


3 


1.3750 


33 


38.0416 


3 


.0^4 


33 


8.4027 


4 


1.9259 


34 


39.9814 


4 


.1234 


34 


8.9197 


5 


2.5231 


35 


41.9675 


5 


.1928 


35 


9.4521 I 


6 


3.1666 


36 


44.0000 


6 


.2778 


36 


10.0000 


7 


3.8564 


37 


46.0787 


7 


.3781 


37 


10.5632 1 


8 


4.5925 


38 


48.2037 


8 


.4938 


38 


11.1419 ; 


9 


5.3750 


39 


50.3750 


9 


.6250 


39 


11.7361 1 


10 


6,2037 


40 


52.5925 


10 


.7716 


40 


12.3456 


11 


7.0787 


41 


54.8564 


11 


.9336 


41 


12.9706 


12 


8.0000 


42 


57.1666 


12 


1.1111 


42 


13.6111 


13 


8.9675 


43 


59.5231 


13 


1.3040 


43 


14.2680 


14 


9.9814 


44 


61.9259 


14 


1.5123 


44 


14.9382 


15 


11.0416 


45 


64.3750 


15 


1.7361 


45 


15.6250 


16 


12.1481 


46 


66.8703 


16 


1.9753 


46 


16.3271 


17 


13.3009 


47 


69.4120 


17 


2.2299 


47 


17.0447 


18 


14.5000 


48 


72.0000 


18 


2.5000 


48 


17.7777 


19 


15.7453 


49 


74.6342 


19 


2.7856 


49 


18.6262 


20 


17.0370 


50 


77.3148 


20 


3.0864 


50 


19.2901 


21 


18.3750 


61 


80.0416 


21 


3.4028 


51 


20.0694 


22 


19.7592 


52 


82.8148 


22 


3.7346 


52 


20.8641 


23 


21.1898 


53 


86.6342 


23 


4.0818 


63 


21.6743 


24 


22.6666 


54 


88.6000 


24 


4.4444 


54 


22.6000 


25 


24.1898 


55 


91.4120 


25 


4.8228 


55 


23.3410 


26 


25.7592 


56 


94.3703 


26 


6.2160 


66 


24.1976 


27 


27.3750 


57 


97.3750 


27 


6.6250 


67 


25.0694 


28 


29.0370 


58 


100.4259 


28 


6.0494 


58 


25.9567 


29 


30.7453 


59 


103.5231 


29 


6.4891 


59 


26.8696 


30 


32.5000 


60 


106.6666 


30 


6.9444 


60 

• 


27.7777 



" ' 


(xtL) 


BASE 21— SLOPE IJ to 1. 


1 


Add. 




Add. 




Deduct. 


if. of 

Ht8. 


Deduct. 


m 




X 




Q 




Q 




1 


.4166 


31 


38.7500 


1 


.0092 


31 


8.8981 


2 


.8888 


32 


40.8888 


2 


.0370 


32 


9.4815 


3 


1.4166 


33 


43.0833 


3 


.0833 


33 


10.0833 


4 


2.0000 


34 


45.3333 


4 


.1480 


34 


10.7037 


5 


2.6388 


35 


47.6388 


5 


.2313 


36 


11.3^25 


6 


3.3333 


36 


50.0000 


6 


.3333 


36 


12.0000 


7 


4.0833 


37 


52.4166 


7 


.4537 


37 


12.6759 


8 


4.8888 


38 


54.8888 


8 


.5926 


38 


13.3703 


9 


5.7500 


39 


67.4166 


9 


.7500 


39 


14.0833 


10 


6.6666 


40 


60.0000 


10 


.9259 


40 


148148 


11 


7.6388 


41 


62.6388 


11 


1.1203 


41 


16.5648 


12 


8.6666 


42 


65.3333 


12 


1.3333 


42 


16.3333 


13 


9.7500 


43 


68.0833 


13 


1.5648 


43 


17.1202 


14 


10.8888 


44 


70.8888 


14 


1.8148 


44 


17.9259 


15 


12.0833 


45 


73.7500 


16 


2.0833 


45 


18.7500 


16 


13.3333 


46 


76.6666 


16 


2.3704 


46 


19.6926 


17 


14.6388 


47 


79.6388 


17 


2.6759 


47 


20.4537 


18 


16.0000 


48 


82.6666 


18 


3.0000 


48 


21.3333 


19 


17.4166 


49 


85.7500 


19 


3.3426 


49 


22.2314 


20 


18.8888 


50 


88.8888 


20 


3.7037 


50 


23.1481 


21 


20.4166 


51 


92.0833 


21 


40833 


61 


24.0833 


22 


22.0000 


52 


95.3333 


22 


4.4815 


62 


26.0370 


23 


23.6388 


53 


98.6388 


23 


48981 


53 


26.0092 


24 


25.3333 


54 


102.0000 


24 


6.3333 


64 


27.0000 


25 


27.0833 


55 


105.4166 


26 


5.7869 


55 


28.0092 


26 


28.8888 


56 


108.8888 


26 


6.2592 


56 


29.0370 


27 


30.7500 


57 


112.4166 


27 


6,7500 


67 


30.0833 


28 


32.6666 


58 


116.0000 


28 


7.2592 


58 


31.1481 


29 


34.6388 


59 


119.6388 


29 


7.7869 


69 


32.2314 


30 


36.6666 

• 


60 


123.3333 


30 


8.3333 


60 


33.3333 



J 





( xvii. ) 




BASE 21--SL0PE If to 1. 


} 


Add. 




A4d. 




• 

Deduct. 


|s 


Deduct. 


ffl 




H 




1 








1 


.4212 


31 


43.1990 


.0108 


31 


10.3811 


2 


.9074 


32 


45.6296 


2 


.0432 


32 


11.0609 


8 


1.4583 


33 


48.1250 


8 


.0972 


83 


11.7688 


4 


2.0740 


34 


60.6851 


4 


.1728 


84 


12.4876 


5 


2.7546 


85 


53.8101 


5 


.2700 


85 


18.2330 


6 


3.6000 


86 


56.0000 


6 


.3889 


86 


14.0000 


7 


4.3101 


37 


68.7546 


7 


.5293 


37 


14.7885 


8 


6.1851 


88 


61.5740 


8 


.6913 


38 


15.6987 


9 


6.1280 


39 


64.4683 


9 


.8760 


39 


16.4306 


10 


7.1296 


46 


67.4074 


10 


1.0802 


40 


17.2839 


11 


8.1990 


41 


70.4212 


11 


1.3071 


41 


18.1589 


12 


9.3338 


42 


73.6000 


12 


1.5555 


42 


19.0555 


13 


10.6324 


43 


76.6435 


18 


1.8256 


48 


19.9747 


14 


11.79^ 


44 


79.8618 


14 


2.1173 


44 


20.9136 


16 


18.1250 


46 


83.1260 


16 


2.4305 


45 


21.8760 


16 


14.5185 


46 


86.4629 


16 


2.7664 


46 


22.8580 


17 


16.9768 


47 


89.8657 


17 


3.1219 


47 


23.8626 


18 


17.6000 


48 


93.3333 


18 


3.6000 


48 


24.8888 


19 


19.0879 


49 


96.8657 


19 


3.8997 


49 


25.9367 


20 


20.7407 


60 


100.4629 


20 


4.3210 


60 


27.0061 


61 


22.4583 


51 


104.1250 


21 


4.7639 


51 


28.0972 


22 


24.2407 


62 


107.8518 


22 


6.^84 


52 


29.2098 


23 


26.0879 


53 


111.6435 


23 


5.7145 


63 


30.3441 


24 


28.0000 


64 


116.6000 


24 


6.2222 


54 


31.6000 


25 


29.9768 


55 


119.4212 


25 


6.7516 


66 


32.6774 


26 


82.0186 


56 


123.4074 


26 


7.3024 


56 


88.8765 


27 


841260 


57 


127.4683 


27 


7.8760 


57 


35.0972 


28 


36.2962 


58 


131.6740 


28 


8.4691 


58 


36.3394 


29 


38.6324 


59 


135.7546 


29 


9.0848 


59 


37.6038 


30 


40.8338 


60 


140.0000 


30 


9.7222 


60 

D 


38.8888 



( xriii. ) 


BASE 21— SLOPE 2 to 1. 


1 


Add. 


1 


Add. 


"a 


Deduct. 




Dedoct. 


1 




» 




a 




Q 




.4259 


31 


47.6481 


1 


.0123 


31 


11.8642 


2 


.9259 


32 


60.3703 


2 


.0494 


32 


12.6419 


3 


1.5000 


33 


63.1666 


3 


.1111 


33 


13.4444 


4 


2.1481 


34 


56.0370 


4 


.1975 


34 


14.2716 


5 


2.8703 


36 


68.9814 


6 


.3086 


36 


16.1234 


6 


3.6666 


36 


62.0000 


6 


.4444 


36 


16.0000 


7 


46370 


37 


65.0925 


7 


.6049 


37 


16.9012 


8 


5.4814 


38 


68.2692 


8 


.7901 


38 


17.8271 


9 


6.6000 


39 


71.6000 


9 


1.0000 


39 


18.7777 


10 


7.5925 


40 


748148 


10 


1.2346 


40 


19.7590 


11 


8.7692 


41 


78.2037 


11 


1.4938 


41 


20.7630 


12 


10.0000 


*2 


81.6666 


12 


1.7778 


42 


21.7777 


13 


11.3148 


43 


85.2037 


13 


2.0864 


43 


22.8271 


14 


12.7037 


44 


88.8148 


14 


2.4197 


44 


23.9012 


15 


14.1666 


46 


92.5000 


16 


2.7778 


45 


26.0000 


16 


15.7037 


46 


96.2592 


16 


3.1606 


46 


26.1234 


17 


17.3148 


47 


100.0925 


17 


3.6679 


47 


27.2716 


18 


19.0000 


48 


104.0000 


18 


40000 


48 


28.4444 


19 


20.7692 


49 


107.9814 


19 


44568 


49 


29.6420 


20 


22.5928 


60 


112.0870 


20 


49382 


50 


30.8642 


21 


246000 


61 


116.1666 


21 


6.4444 


61 


32.1111 


22 


26.4814 


62 


120.3703 


22 


5.9753 


62 


33.3827 


23 


28.5370 


53 


1246481 


23 


6.5309 


63 


34.6790 


24 


30.6666 


64 


129.0000 


24 


7.1111 


64 


36.0000 


25 


32.8703 


55 


133.4259 


25 


7.7160 


66 


37.3466 


26 


35.1481 


56 


137.9269 


26 


8.3467 


66 


38.7160 


27 


37.6000 


67 


142.5000 


27 


9.0000 


67 


40.1111 


28 


39.9259 


58 


147.1481 


28 


9.6790 


68 


41.6308 


29 


42.^69 


69 


151.8703 


29 


10.3827 


59 


42.9763 


30 


46.0000 

4 


60 


156.6666 


30 


11.1111 60 


44.4444 



- 




( xix. ) 




BASE 21 SLOPE ^ to 1. 


t 


A4d. 


1 


Add. 


Dif.of 


Deduct. 


Dif.of 
Ht(. 


Deduct. 


1 


.4351 


31 


56.5462 


1 


.0154 


31 


148302 


2 


.9^9 


32 


59.8518 


2 


.0617 


32 


16.8018 


3 


1.5833 


33 


63.2500 


3 


.1389 


33 


1&8055 


4 


2.2962 


34 


66.7407 


4 


.2468 


34 


17.8896 


5 


aioi8 


35 


70.3240 


5 


.3857 


36 


18.9043 


6 


40000 


36 


740000 


6 


.6666 


36 


20.0000 


7 


49907 


37 


77.7685 


7 


.7561 


37 


21.1265 


8 


6X)740 


38 


81.6296 


8 


.9876 


38 


22.2839 


9 


7.2500 


39 


85.5833 


9 


1.2600 


39 


23.4722 


10 


a5185 


40 


89.6296 


10 


1.6432 


40 


24.6913 


11 


9.8796 


41 


98.7685 


11 


1.8673 


41 


26.9413 


12 


11.2962 


42 


98.0000 


12 


2.2222 


42 


27.22J^ 


13 


12.8796 


43 


102.3240 


13 


.2.6080 


43 


2a6360 


14 


14.5185 


44 


106.7407 


14 


3.0247 


44 


29.8765 


IS 


1&2500 


45 


111.2500 


15 


3.4722 


45 


81.2600 


16 


18.0740 


46 


115.8618 


16 


3.9606 


46 


32.6543 


17 


19.9907 


47 


120.546i 


17 


44699 


47 


84.0895 


18 


22.0000 


48 


125.3333 


18 


5.0000 


48 


35.5566 


19 


241018 


49 


130.2129 


19 


6.6710 


49 


37.0624 


20 


26.2962 


50 


135.1861 


20 


6.1728 


50 


38..5802 


21 


28.5833 


51 


140.2600 


21 


6.8065 


51 


40.1388 


22 


30.9629 


52 


146.4074 


22 


7.4691 


52 


41.7283 


23 


33.4351 


53 


160.6674 


23 


8.1636 


53 


43.8487 


24 


36.0000 


54 


156.0000 


24 


8.8889 


54 


46.0000 


25 


88.6574 


65 


161.4361 


25 


9.6466 


65 


46.6820 


26 


41.4074 


56 


166.9629 


26 


10.4821 


66 


48.3960 


27 


442500 


67 


172.6833 


27 


11.2500 


57 


50.1388 


28 


47.1851 


58 


178.2962 


28 


12.0587 


58 


51.9135 


29 


50.2129 


59 


1841018 


29 


12.9782 


59 


53.7191 


30 


63.3333 


60 


190.0000 


30 


13.8889 


60 


66.5565 



(zx.) 
BASE 21— SLOPE 3 to 1. 



» 



1 
2 
3 
4 
& 

6 
7 
8 
9 
10 

11 
12 

13 
14 
15 

16 
17 
18 
19 
20 

21 
22 
23 
24 
25 

26 
27 
28 
29 
30 



Add. 



AAAA 

• X J. X X 

1.0000 
1.6666 
2.4444 
a333d 

4.SS33 
5.4444 
6.6666 
8.0000 
9.4444 

ll.OOOQ 
12.6666 

UA4A4 
• XX X X 

16.3333 
18.3333 

20.4444 
22.6666 
25.0000 
27.4444 
30.0000 

32.6666 
35.4444 
38.3333 
41.3333 
44.4444 

47.6666 
51.0000 
54.4444 
58.0000 
61.6666 






31 
32 
33 
34 
35 

36 
37 
38 
39 

4& 

41 
42 
43 
44 
45 

46 
47 

48^ 
49 
50 

51 
52 
53 
54 
55 

56 
57 

58 
59 
60 



Add. 



65.4444 
69.3333 
73;3333 
77.4444 
81.6666 

86.0000 
90.4444 
95.0000 
99.6666 
104.4444 

109.3333 
114.3333 
119.4444 
124.6666 
130.0000 

135.4444 
141.0000 
146.6666 
130.0000 
135.4444 

141.0000 
146.6666 
152.4444 
158.3333 
164.3333 

170.4444 
202.6666 
209.4444 
216.3333 
223.3333 






1 

2 
3 
4 
6 



Deduct. 



.0185 
.0740 
.1667 
.2961 
.4628 



".a 



g 



Deduct. 



6 


.6667 


7 


.9074 


8 


1.1852 





1.5000 


10 


1.8518 


11 


2.2407 


12 


2.6667 


13 


3.129& 


14 


3.6296 


16 


4.1667 


16 


4.7407 


17 


5.3516 


18 


6.0000^ 


1» 


a6852 


20 


7.4074 


21 


8.1667 


22 


8.9629 


23 


9.7962 


24 


10.6667 


25 


11.5741 


26 


12.5184 


27 


13.5000 


28 


14.5185 


29 


15.5739 


30 


16.6667 



m 17.7963 

32 18.9630 

33 20.1666 

34 21.4074 

35 22.6851 

36 24.0000 

37 25.3518 

38 26.7407 

39 28.1666 

40 29.6296 

41 31.1296 

42 32.6666 

43 34.2405 

44 35.8518 

45 37.5000 

46 39.1851 

47 40.9074 

48 42.6666 

49 44.4629 

50 46.2963 

51 49.166ft 

52 60.0740 

53 52.0185 

54 54.0000 

55 56.0184 

m 58.0640 

57 60.1666 

58 62.2962 

59 64.4629 

60 66.666ft 



; 


(ixi.) 


BASE 22— SLOPE i to 1. 


^ 1 

\ a 


Add. 


i 

a 

31 


Add. 


ga 
1 


Deduct. 


2^ 

ga 

31 


Deduct, 


1 

1 


.4120 


17.0787 


.0016 


1.4830 


2 


.8333 


82 


17.7777 


2 


.0062 


82 


1.5802 


3 


1.2638 


83 


18.4861 


3 


.0139 


33 


1.6805 


4 


1.7037 


34 


19.2087 


4 


.0246 


34 


1.7839 


5 


2.1627 


85 


19.9305 


6 


.0385 


36 


1.8904 


6 


2.6111 


36 


20.6666 


6 


.0565 


86 


2.0000 


7 


3.0787 


37 


21.4120 


7 


.0766 


87 


2.1126 


8 


3.5555 


38 


22.1666 


8 


.0988 


38 


2.2284 


9 


40416 


89 


22.9306 


9 


.1260 


39 


2.8472 


10 


4.5370 


40 


28.7037 


10 


.1643 


40 


2.4691 


11 


5.0416 


41 


24.4861 


11 


.1867 


41 


2.5941 


12 


5.5555 


42 


26.2777 


12 


.2222 


42 


2.7222 


13 


6.0787 


43 


26.0787 


18 


.2608 


48 


2.8646 


14 


6.6111 


44 


26.8888 


14 


.3026 


44 


2.9876 


15 


7.1527 


45 


27.7088 


15 


.8472 


45 


3.1260 


16 


7.7087 


46 


28.6870 


16 


.3961 


46 


8.2654 


17 


8.2638 


47 


29.8760 


17 


.4460 


47 


3.4089 


18 


8.8333 


48 


30.2222 


18 


.5000 


48 


3.5555 


19 


9.4120 


49 


31.0787 


19 


.6671 


49 


3.7052 


20 


10.0000 


60 


31.9444 


20 


.6173 


60 


3.8580 


21 


10.5972 


51 


32.8194 


21 


.6806 


61 


4.0139 


22 


11.2037 


52 


83.7037 


22 


.7469 


62 


4.1728 


23 


11.8194 


58 


84.5972 


23 


.8163 


53 


4.3349 


24 


12.4444 


54 


35.5000 


24 


.8889 


64 


4.6000 


25 


13.0787 


55 


36.4120 


25 


.9647 


65 


4.6682 


26 


13.7222 


66 


37.3333 


26 


1.0432 


66 


4.8396 


27 


14.3750 


67 


38.2638 


27 


1.1250 


67 


5.0189 


28 


16.0370 


58 


39.2037 


28 


1.2099 


58 


6.1913 


29 


15.7088 


59 


40.1527 


29 


1.2978 


69 


6.3719 


30 


16.3888 60 


41.1111 


30 


1.3889 


60 


5.5665 





( ""• ) 


BASE 22— SLOPE J to 1. | 


1 

a 
1 


Add. 


1 

a 


Add. 


Dif. of 
Utf. 


Deduct. 


Dif. of 
Hts. 


Dedoot. 


.4166 


31 


21.5277 


1 


.0031 


31 


2.9680 


2 


.8518 


32 


22.6185 


2 


.0123 


32 


3.1605 


3 


1.3055 


33 


23.6277 


3 


.0278 


33 


36111 


4 


1.7777 


34 


24.5555 


4 


.0494 


34 


3.5679 


6 


2.2685 


35 


26.6018 


5 


.0772 


36 


3.7808 


6 


2.7777 


36 


26.6666 


6 


.1111 


36 


4.0000 


7 


3.3055 


37 


27.7600 


7 


.1512 


37 


4.2263 


8 


3.8518 


38 


28.8518 


8 


.1976 


38 


4.4668 


9 


4.4166 


39 


29.9722 


9 


.2500 


39 


4.6944 


10 


6.0000 


40 


31.1111 


10 


.3086 


40 


4.9383 


11 


5.6018 


41 


32.2686 


11 


.3734 


41 


5.1883 


12 


6.2222 


42 


33.4444 


12 


.4444 


42 


5.4444 


13 


6.8611 


43 


34.6388 


13 


.5216 


43 


5.7067 


14 


7.5185 


44 


35.8618 


14 


.6049 


44 


5.9753 


16 


8.1944 


45 


37.0833 


15 


.6944 


45 


6.2500 


16 


8.8888 


46 


38.3333 


16 


.7901 


46 


6.5308 


17 


9.6018 


47 


39.6018 


17 


.8920 


47 


6.8179 


18 


10.3333 


48 


40.8888 


18 


1.0000 


48 


7.1111 


19 


11.0833 


49 


42.1944 


19 


1.1142 


49 


7.4104 


20 


11.8518 


50 


43.6185 


20 


1.2346 


50 


7.7160 


21 


12.6388 


51 


44.8611 


21 


1.3611 


51 


8.0277 


22 


13.4444 


52 


46.2222 


22 


1.4938 


62 


8.3456 


23 


14.2685 


53 


47.6018 


23 


1.6327 


63 


8.6697 


24 


15.1111 


54 


49.0000 


24 


1.7778 


54 


9.0000 


25 


15.9722 


55 


50.4166 


25 


1.9296 


65 


9.3364 


26 


16.8518 


56 


61.8518 


26 


2.0864 


66 


9.6790 


27 


17.7500 


57 


63.3055 


27 


2.2500 


57 


10.0277 


28 


18.6666 


58 


64.7777 


28 


2.4197 


58 


10.3827 


29 


19.6018 


59 


66.2685 


29 


2.6956 


59 


10.7438 


30 20.55551 


60 


67.7777 


30 


2.7778 


60 


11.1111 



J 



( xxiii. ) 
BASE 22— SLOPE | to 1. 



•a 

•s 


Add. 


<fc» 


Add. 


mi 


Deduct. 




Deduct. 


as 




a 




1 




Q*" 




1 


.42r2 


31 


26.9768 


.0046 


31 


4.4490 


2 


.8703 


32 


27.2592 


2 


.0186 


32 


4.7407 


3 


1.3472 


33 


28.5694 


3 


.0416 


33 


5.0416 


4 


1.8618 


34 


29.9074 


4 


.0740 


34 


5.3518 


5 


2.3842 


36 


31.2731 


6 


.1167 


36 


66712 


6 


2.9444 


36 


32.6666 


6 


.1667 


36 


6.0000 


7 


3.6324 


37 


34.0879 


7 


.2268 


37 


63379 


8 


4.1481 


38 


36.6370 


8 


.2963 


38 


6.6851 


9 


4.7916 


39 


37.0138 


9 


.3750 


39 


7.0416 


10 


6.4629 


40 


38.6186 


10 


.4630 


40 


7.4074 


11 


6.1620 


41 


40.0609 


11 


.5602 


41 


7.7824 


12 


6.8888 


42 


41.6111 


12 


.6667 


42 


8.1666 


13 


7.6436 


43 


43.1990 


13 


.7824 


43 


8.5612 


14 


8.4269 


44 


44.8148 


14 


.9074 


44 


89629 


15 


9.2361 


46 


46.4583 


16 


1.0417 


45 


9.3750 


16 


10.0740 


46 


48.1296 


16 


1.1852 


46 


9.7962 


17 


10.9398 


47 


49.8287 


17 


1.3379 


47 


10.2268 


18 


11.8333 


48 


51.6555 


18 


1.5000 


48 


10.6666 


19 


12.7509 


49 


63.3101 


19 


1.6713 


49 


11.1167 


20 


13.7037 


50 


56.0925 


20 


1.8518 


60 


11.6740 


21 


14.6806 


61 


66.9027 


21 


2.0417 


51 


12.0416 


22 


16.6851 


62 


58.7407 


22 


2.2407 


62 


12.6186 


32 


16.7175 


63 


60.6064 


23 


2.4491 


63 


13.0046 


24 


17.7777 


54 


62.6000 


24 


2.6667 


64 


13.5000 


26 


ia8667 


55 


64.4212 


26 


2.8936 


66 


14.0046 


26 


19.9814 


66 


66.3703 


26 


3.1296 


56 


14.5185 


27 


21.1260 


67 


68.3472 


27 


3.3760 


67 


15.0416 


28 


$^.2962 


68 


70.3518 


28 


3.6296 


68 


15.6740 


29 


23.^63 


69 


72.3842 


29 


3.8936 


59 


16.1157 


30 


24.7222 


60 


74.4444 


30 


4.1667 


60 


1&6666 





( xxiv. ) 


BASE 22— SLOPE 1 to 1. f| 


i 


Add. 


1 


Add. 




Deduct, 




Deduct. 


» 




X 




1 




o 


; 


1 


.4259 


31 


30.4269 


.0062 


3i 


5.9321 


2 


.8888 


32 


32.0000 


2 


.0247 


32 


6.3210 • 


3 


1.3888 


33 


33.6111 


3 


.0656 


33 


6.7222 : 


4 


1.9259 


34 


36.2692 


4 


.0988 


34 


7.1358 i 


5 


2.5000 


36 


36.9444 


6 


.1643 


36 


7.6617 


6 


3.1111 


36 


38.6666 


6 


.2222 


36 


8.0000 


7 


3.7692 


37 


40.^69 


7 


.3026 


37 


8.4506 ; 


8 


4.4444 


38 


42.2222 


e 


.3951 


38 


8.9136 ' 


9 


5.1666 


89 


44.0655 


9 


.5000 


39 


9.3888 


10 


5.9259 


40 


45.9259 


10 


.6173 


40 


9.8766 


11 


6.7222 


41 


47.8333 


11 


.7469 


41 


10.3765 


12 


7.5555 


42 


49.7777 


12 


.8889 


42 


10.8888 


13 


8.4259 


43 


51.7692 


13 


1.0432 


43 


11.4136 

1 


14 


9.3333 


44 


53.7777 


14 


1.2099 


44 


11.9506 


15 


10.2777 


45 


65.8333 


15 


1.3889 


46 


12.5000 


16 


11.2592 


46 


67.9269 


16 


1.5802 


46 


13.0617 


17 


12.2777 


47 


60.0556 


17 


1.7839 


47 


13.6358 


18 


13.3333 


48 


62.2222 


18 


2.0000 


48 


14.2222 


19 


14.4259 


49 


64.4959 


19 


2.2284 


49 


14.8209 


20 


15.5555 


50 


66.6666 


20 


2.4691 


60 


15.4321 


21 


16.7222 


51 


68.9444 


21 


2.7222 


61 


16.0556 


22 


17.9259 


52 


71.2592 


22 


2.9876 


52 


16.6913 


23 


19.1666 


53 


73.6111 


23 


3.2654 


63 


17.3396 


24 


20.4444 


64 


76.0000 


24 


3.5555 


64 


18.0000 


25 


21.7692 


56 


78.4269 


25 


3.8530 


55 


18.6728 


26 


2ailll 


56 


80.8888 


26 


4.1728 


56 


19.3580 


27 


24.6000 


57 


83.3888 


27 


4.6000 


57 


20.0655 


28 


25.9269 


58 


«6.9259 


28 


4.8395 


58 


20.7654 


29 


27.3888 


69 


88.6000 


29 


5.1913 


69 


21.4876 i 


30 


28.8888 


60 


91.1111 


30 


6.5565 


60 


22.2222 

- « 



.- , - ■ ' 

( ^*^* ) 


BASE 22— SLOPE IJ to 1. 


■g, 
% 


Add. 


1 


Add. 


s 

ti m Deduct. 


O • 


Sednct. 


X 

1 




» 




Q'^ 




o* 




.4305 


31 


34.8760 


I 


.0077 


31 


.7.4151 


2 


.9074 


32 


36.7407 


2 


.0309 


32 


7.9012 


3 


1.4305 


33 


38.6527 


3 


.0694 


33 


8.4027 


4 


2.0000 


34 


40.6111 


4 


.1234 


34 


8.9197 


5 


2.6167 


35 


42.6167 


6 


.1928 


36 


d.4521 


6 


3.2777 


36 


44.6666 


6 


.2778 


36 


laoooo 


7 


3.9861 


37 


46.7638 


7 


.3781 


37 


10.5632 


« 


4.7407 


38 


48.9074 


8 


.4938 


38 


11.1419 


9 


6.6416 


39 


51.0972 


9 


.6250 


39 


11.7361 


10 


&3888 


40 


63.3333 


10 


.7716 


40 


12.3456 


11 


7.2824 


41 


65.6167 


11 


.9336 


41 


12.9706 


12 


8.2222 


42 


57.9444 


12 


1.1111 


42 


13.6111 


13 


9.2083 


43 


60.3194 


13 


1.3040 


43 


14.2680 


14 


10.2407 


44 


62.7407 


14 


1.6123 


44 


149382 


15 


11.3194 


45 


66.2083 


16 


1.7361 


45 


15.6260 


16 


12.4444 


46 


67.7222 


16 


1.9763 


46 


16.3271 


17 


13.6157 


47 


70.2824 


17 


2.2299 


47 


17.0447 


18 


14.8333 


48 


72.8888 


18 


2.5000 


48 


17.7777 


19 


16.0972 


49 


75.6416 


19 


2.7865 


49 


18.5262 


20 


17.4074 


60 


78.2407 


20 


3.0864 


60 


19.2901 


21 


18.7638 


51 


80.9861 


21 


a4028 


61 


20.0694 


22 


20.1666 


52 


83.7777 


22 


3.7346 


52 


20.8641 


23 


21.6167 


63 


86.6167 


23 


4.0818 


53 


21.6743 


24 


23.1111 


64 


89.5000 


24 


4.4444 


54 


22.6000 


26 


24.6527 


56 


92.4306 


2S 


4.8228 


65 


23.3410 


26 


26.2407 


66 


95.4074 


26 


5.2160 


66 


24.1976 


27 


27.8750 


57 


98.4305 


27 


5.6250 


57 


26.ce94 


28 


29:5655 


58 


101.6000 


28 


6.0494 


68 


25;9567 


29 


31.2824 


59 


104.6157 


29 


6.4891 


69 


26.8595 


30 


3a0566 


60 


107.7777 


30 6.94441 


60 

E 


iW.7777 





( xxvi. ) 


BASE 22-8LOPE IJ to 1. 

1 


1 


Add. 


1 


Add. 




Deduct. 




Deduct. 


ffi 




a 




1 




o*" 




1 


.4351 


31 


39.3240 


.0092 


31 


8.8981 


2 


.9259 


32 


41.4814 


2 


.0370 


32 


9.4816 


8 


1.4722 


33 


43.6944 


3 


.0633 


33 


10.0833 


4 


2.0740 


34 


46.9629 


4 


.1480 


34 


10.7037 


6 


2.7314 


35 


48.2870 


6 


.2313 


35 


11.3436 


6 


3.4444 


36 


50.6666 


6 


.3333 


36 


12.0000 


7 


42129 


37 


53.1018 


7 


.4537 


37 


12.6759 


8 


5.0370 


38 


66.5925 


8 


.5926 


38 


13.3708 


9 


5.9166 


39 


68.1388 


9 


.7500 


89 


14.0833 


10 


a851R 


40 


60.7407 


10 


.9259 


40 


14.8148 


11 


7.8425 


41 


63.3081 


11 


1.1203 


41 


15.6648 


12 


8.8888 


4a, 


66.1111 


12 


1.3333 


42 


16.3833 


13 


9.9907 


43 


68.8796 


13 


1.5648 


48 


17.1202 


14 


11.1481 


44 


71.7037 


14 


1.8148 


44 


17.9269 


15 


12.8611 


45 


74.5833 


15 


2.0833 


45 


18.7600 


16 


13.6296 


46 


77.6186 


16 


2.3704 


46 


19.5926 


17 


14.9537 


47 


80.5092 


17 


2.6769 


47 


20.4637 


18 


1&3333 


48 


83.5566 


18 


3.0000 


48 


21.3333 


19 


17.7685 


49 


86.6674 


19 


3.3426 


49 


22.2314 


20 


19.2592 


50 


89.8148 


20 


3.7037 


60 


23.1481 


21 


20.8055 


61 


93.0277 


21 


4.0833 


51 


24.0833 


22 


22.4074 


52 


96.2962 


22 


4.4816 


62 


25.0870 


23 


24.0648 


53 


99.6203 


23 


4.8981 


53 


26.0092 


24 


25.7777 


64 


103.0000 


24 


6.3333 


64 


27.0000 


25 


27.6462 


55 


106.4361 


25 


5.7869 


65 


28.0092 


26 


29.3703 


66 


109.9269 


26 


6.2692 


56 


29.0370 


27 


31.2600 


57 


113.4722 


27 


6.7600 


67 


3J.0833 


28 


33.1861 


58 


117.0740 


28 


7.2692 


68 


31.1481 


29 


36.1759 69 


120.7314 


29 


7.7869 


69 32.2314 1 


30 


37.2222 60 


124.4444 


30 


8.3333 60 33.3333 



( xxTii. ) 


BASE 22— SLOPE If to 1. 


•a, 

•s 


Add. 


i 


Add. 




Deduct. 


•a 


Dednet. 


!S 




X 
31 




& >*♦ 




Q* 




1 


.4398 


43.7731 


1 


.0108 


81 


10.3811 


2 


.9444 


32 


46.2222 


2 


.0432 


82 


11.0609 


8 


1.5138 


33 


48.7361 


3 


.0972 


88 


11.7638 


4 


2.1481 


34 


61.3148 


4 


.1728 


34 


12.4876 


6 


2.8472 


36 


53.9583 


5 


.2700 


36 


13.2330 


6 


3.6111 


36 


66.6666 


6 


.3889 


86 


140000 


7 


4.4398 


37 


69.4398 


7 


.5293 


87 


147886 


8 


5.3333 


38 


62.2777 


8 


.6913 


38 


16.5987 


9 


6.2916 


39 


65.1805 


9 


.8750 


39 


16.43(»5 


10 


7.3148 


40 


68.1481 


10 


1.0802 


40 


17.2839 


11 


8.4027 


41 


71.1805 


11 


1.3071 


41 


18.1689 


12 


9.5556 


42 


74.2777 


12 


1.6666 


42 


19.0555 


13 


10.7731 


43 


77.4398 


13 


1.8266 


43 


19.9747 


14 


12.0565 


44 


80.6666 


14 


2.1173 


44 


20.9136 


15 


13.4027 


46 


83.9683 


16 


2.4a06 


45 


21.8750 


16 


148148 


46 


87.3148 


16 


2.7654 


46 


22.8680 


17 


16.2916 


47 


90.7361 


17 


3.1219 


47 


28.8626 


18 


17.8333 


48 


94.2222 


18 


3.5000 


48 


24.8888 


19 


19.4398 


49 


97.7731 


19 


8.8997 


49 


26.93(»7 


20 


21.1111 


50 


101.3888 


20 


43210 


60 


27.0061 


21 


22.8472 


61 


105.0694 


21 


47639 


51 


28.0972 


22 


24.6481 


52 


108.8148 


22 


5.2284 


62 


29.2098 


i3 


26.6137 


53 


112.6260 


23 


6.7146 


53 


30.3441 


24 


28.4444 


54 


116.5000 


24 


6.2222 


64 


31.6000 


26 


30.4398 


56 


120.4398 


25 


6.7516 


55 


32.6774 


26 


32.6000 


56 


124.4444 


26 


7.3024 


56 


83.8766 


27 


846250 


57 


128.5138 


27 


7.8750 


57 


35.0972 


28 


36.8148 


58 


132.6481 


28 


8.4691 


68 


36.3394 


29 


39.0694 


69 


136.8472 


29 


9.0848 


69 


37.6033 


SO 


41.8888 


60 


141.1111 


30 


9.7222 


60 


88.8888 





( xxviii. ) 


. 




BASE 22 SLOPE 2 to 1. 






4^ 


Add. 


1 


Add. 


if. of 


Deduct. 


° s 

■SB' Deduct. 




• 


n 
1 




X 

31 




1 




P"^ 








AAA4 

• Jl X X X 


48.2222 


.0123 


31 


11.8642 






2 


.9629 


32 


60.9629 


2 


.0494 


32 


12.6419 


' 




3 


1.6666 


33 


63.7777 


3 


.1111 


33 


13.4444 






4 


2.2222 


34 


66.6666 


4 


.1976 


34 


14.2716 






5 


2.9629 


36 


69.6296 


6 


.3086 


36 


15.1234 






6 


3.7777 


36 


62.6666 


6 


AAAA 

• M X'X'X 


36 


16.0000 






7 


4.6666 


37 


66.7777 


7 


.6049 


37 


16.9012 






, 8 


6.6296 


38 


68.9629 


8 


.7901 


38 


17.8271 






9 


6.6666 


39 


72.2222 


9 


1.0000 


39 


18.7777 






10 


7.7777 


40 


76.6566 


10 


1.2346 


40 


19.7630 






11 


8.9629 


41 


78.9629 


11 


1.4938 


41 


20.7530 






12 


10.2222 


42 


82.4444 


12 


1.7778 


42 


21.7777 






13 


11.6666 


43 


86.0000 


13 


2.0864 


43 


22.8271 






14 


12.9629 


41 


89.6296 


14 


2.4197 


44 


23.9012 






16 


14.4444 


46 


93.3333 


16 


2.7778 


46 


26.0000 






16 


16.0000 


46 


97.1111 


16 


3.1605 


46 


26.1234 






17 


17.6296 


47 


100.9629 


17 


3.6679 


47 


27.2716 






18 


19.3333 


48 


104.8888 


18 


4.0000 


48 


28.4444 






19 


21.1111 


49 


108.8888 


19 


4.4668 


49 


29.6420 






20 


22.9629 


60 


112.9629 


20 


4.9382 


60 


80.8642 






21 


24.8888 


61 


117.1111 


21 


6.4444 


51 


32.1111 






: 22 


26.8888 


62 


121.3333 


22 


6.9753 


52 


33.3827 






23 


28.9629 


63 


126.6296 


23 


6.6309 


53 


34.6790 






24 


31.1111 


64 


130.0000 


24 


7.1111 


64 


36.0000 


9 




26 


33.3333 


66 


134.4444 


26 


7,7160 


56 


37.3466 






26 


36.6296 


66 


138.9629 


26 


8.3467 


56 


38.7160 






27 


38.0000 


67 


143.6666 


27 


9.0000 


67 


40.1111 






28 


40.4444 


68 


148.2222 


28 


9.6790 


68 


41.5308 






29 


42.9629 


69 


162.9629 


29 


10.3827 


69 


42.9753 






30 


46.66661 60 1 167.7777 


30 


11.1111 


60 '44.4444 


r. 



( xxix. ) 


BASE 22— SLOPE 2J to 1. 


•4J 

•a. 




t 




■Si 




'Sa 




•s 


Add. 


•s 


Add. 


»3ffi 


Deduct. 


;ss 


Dedoct. 


X 




!S 




Q 




Q 




1 


.4536 


31 


57.1203 


1 


.0164 


31 


14.8302 


2 


1.000«> 


32 


60.4444 


2 


.0617 


32 


15.8015 


3 


1.6388 


33 


63.8611 


3 


.1389 


33 


16.8055 


4 


2.3704 


34 


67.3703 


4 


.2468 


34 


17.8395 


5 


3.1944 


35 


70.9722 


6 


.3867 


36 


18.9043 


6 


4.1111 


36 


74.6666 


6 


.5566 


36 


20.0000 


7 


5.1202 


37 


78.4637 


7 


.7561 


37 


21.1265 


8 


6.2222 


38 


82.3333 


8 


.9876 


38 


22.2839 


9 


7.4166 


39 


86.3055 


9 


1.2600 


39 


23.4722 


10 


8.7036 


40 


90.3704 


10 


1.6432 


40 


24.6913 


11 


10.0833 


41 


94.5277 


11 


1.8873 


41 


25.9413 


12 


11.6556 


42 


98.7777 


12 


9.2222 


42 


27.2222 


13 


13.1203 


43 


103.1203 


13 


2.6080 


43 


28.6360 


14 


14.7777 


44 


107.6665 


14 


3.0247 


44 


29.8765 


15 


17.6278 


45 


112.0633 


16 


3.4722 


46 


31.2600 


16 


18.3704 


46 


116.7037 


16 


3.9506 


46 


32.6643 


17 


20.3055 


47 


121.4166 


17 


4.4599 


47 


34.0895 


18 


22.3338 


48 


126.2222 


18 


5.0000 


48 


36.6656 


19 


24.4537 


49 


131.1203 


19 


6.6710 


49 


37.0624 


20 


26.6666 


50 


136.1111 


20 


6.1728 


50 


38.6802 


21 


28.9722 


61 


141.1944 


21 


6.8066 


61 


40.1388 


22 


31.3703 


62 


146.3703 


22 


7.4691 


62 


41.7283 


23 


33.8611 


53 


151.6388 


23 


8.1636 


63 


43.3487 


24 


36.4444 


54 


167.0000 


24 


8.8889 


64 


45.0000 


25 


39.1203 


65 


162.4536 


26 


9.6465 


66 


46.6820 


26 


41.8888 


56 


168.0000 


26 


10.4321 


66 


48.3950 


27 


44.7500 


67 


174.6389 


27 


11.2600 


57 


50.1388 


28 


47.7036 


58 


179.3703 


28 


12.0587 


58 


51.9135 


29 


60.7600 


59 


186.1944 


29 


12.9782 


59 


53.7191 


30 


53.8888 


60 


191.1111 


30 


13.8889 


60 


55.5655 



■1 




(xxx.) 






BASE 22— SLOPE 3 to 1. 




Add. 


1 

a 


Add. 


Dif.of 
Bti. 


Deduct. 


Dif.of 
Hts. 


Deduct. 


1 


.4628 


31 


66.0185 


1 


.0186 


31 


17.7963 


2 


1.0370 


32 


69.9259 


2 


.0740 


32 


18.9630 


3 


1.7222 


33 


73.9444 


3 


.1667 


33 


20.1666 


4 


2.6185 


34 


78.0740 


4 


.2961 


34 


21.4074 


5 


3.^59 


35 


82.3148 


6 


.4628 


36 


22.6851 


6 


44444 


36 


8&6666 


6 


.6667 


36 


240000 


7 


5.6740 


37 


91.1296 


7 


.9074 


37 


25.3618 


8 


6.8148 


38 


95.7037 


8 


1.1862 


38 


26.7407 


9 


8.1666 


39 


100.3888 


9 


1.5000 


39 


28.1666 


10 


9.6296 


40 


106.1852 


10 


1.8518 


40 


29.6296 


11 


11.2037 


41 


110.0926 


11 


2.2407 


41 


31.1296 


12 


12.8888 


42 


115.1111 


12 


2.6667 


^ 


32.6666 


13 


14.6852 


43 


120.2407 


13 


3.1296 


43 


34.2405 


14 


1&5925 


44 


125.4813 


14 


3.6296 


44 


35.8518 


15 


19.6111 


45 


130.8333 


16 


4.1667 


45 


37.5000 


16 


20.7407 


46 


136.2962 


16 


4.7407 


46 


39.1861 


17 


1^2.9813 


47 


141.8702 


17 


6.3518 


47 


40.9074 


18 


25.3333 


48 


147.5656 


18 


6.0000 


48 


42.6666 


19 


27.7962 


49 


163.3518 


19 


6.6862 


49 


44.4^29 


20 


30.3703 


50 


159.2593 


20 


7.4074 


50 


46.2963 


21 


33.0555 


51 


165.2777 


21 


8.1667 


61 


49.1666 


22 


35.8518 


62 


171.4073 


22 


8.9629 


62 


50.0740 


23 


38.7592 


63 


177.6480 


23 


9.79®2 


53 


62.0185 


24 


41.7777 


54' 


184.0000 


24 


10.6667 


64 


640000 


25 


44.9073 


55 


190.4628 


25 


11.5741 


55 


66.0184 


26 


48.1480 


56 


197.0370 


26 


12.5184 


56 


58.0640 


27 


51.5000 


57 


204.7222 


27 


13.5000 


57 


60.1666 


28 


54.9628 


58 


210.5185 


28 


145186 


58 


62.2962 


29 


68.5370 


59 


217.4258 


29 


15.6739 


69 


644629 


30 


62.2222 


60 


224.4444 


30 


16.6667 


60 


66.6666 



( xxxi. ) 
BASE 23— SLOPE J to 1. 



1 

n 


Add. 


1 

n 


Add. 


Dif. of 
Hts. 


• 

Deduct. 


Dif. of 
HU. 


Deduct. 


1 


.4305 


31 


17.6627 


1 


.0015 


81 


1.4830 


2 


.8708 


32 


18.3708 


2 


.00^ 


32 


1.6802 


3 


1.3194 


33 


19.0972 


3 


.0139 


83 


1.6806 


4 


1.7777 


34 


19.8333 


4 


.0246 


34 


1.7839 


5 


2.2453 


35 


20.6787 


5 


.0885 


36 


1.8904 


6 


2.7222 


36 


21.3833 


6 


.0555 


36 


2.0000 


7 


3.2083 


87 


22.0972 


7 


.0766 


37 


2.1126 


8 


8.7037 


88 


22.8708 


8 


.0988 


38 


2.2284 


9 


4.2063 


39 


23.6627 


9 


.1250 


39 


2.3472 


10 


472SS2 


40 


244444 


10 


.1543 


40 


2.4691 


11 


5.2453 


41 


26.2453 


11 


.1867 


41 


2.6941 


12 


5.7777 


42 


26.0555 


12 


.2222 


42 


2.7222 


13 


6.3194 


43 


26.8760 


13 


.2608 


43 


2.8646 


14 


6.8708 


44 


27.7037 


14 


.3025 


44 


2.9876 


15 


7.4305 


45 


28.5146 


15 


.3472 


45 


3.1260 


16 


8.0000 


46 


29.3888 


16 


.8951 


46 


3.2654 


17 


8.6787 


47 


80.2463 


17 


.4460 


47 


8.4089 


18 


9.1666 


48 


"31.1111 


18 


.6000 


48 


a5665 


19 


9.7638 


49 


81.9861 


19 


.5671 


49 


3.7052 


20 


10.3703 


50 


82.8708 


20 


.6173 


60 


3.8680 


21 


10.9861 


51 


33.7638 


21 


.6805 


61 


40139 


22 


11.6111 


52 


34.6666 


22 


.7469 


62 


41728 


23 


12.2453 


53 


36.5787 


23 


.8163 


63 


4.33^ 


24 


12.8888 


54 


86.6000 


24 


.8889 


54 


4.6000 


25 


13.5416 


55 


87.4305 


26 


.9647 


56 


46682 


26 


14.2037 


56 


38.3703 


26 


1.0432 


56 


48395 


27 


14.8760 


67 


89.8194 


27 


1.1250 


67 


5.0189 


28 


15.5655 


68 


40.2777 


28 


1.2099 


58 


6.1913 


29 


16.2453 


69 


41.2458 


29 


1.2978 


59 


5.3719 


30 


16.9444 

1 


60 


42.2222 


30 


1.3889 


60 


6.5566 





( xxxii. ) 


BASE 23— SLOPE i to 1. 


1 


Add. 


1 
•s 


Add. 




Deduct. 


s a 


Deduct. 


1 




a 




Q 




o 
31 




.4351 


31 


22.1018 


1 


.0031 


2.9660 


2 


.8888 


32 


23.1111 


2 


.0123 


32 


3.1605 


3 


1.3611 


33 


24.1388 


3 


.0278 


33 


3.6111 


4 


1.8518 


34 


25.1851 


4 


.0494 


34 


3.5679 


6 


2.3610 


35 


2&2500 


6 


.0772 


35 


3.7808 

1 


6 


2.8888 


36 


27.3333 


6 


.1111 


36 


4.0000 


7 


3.4351 


37 


28.4351 


7 


.1512 


37 


4.2263 


8 


4.0000 


38 


29.5555 


8 


.1976 


38 


4.4568 


9 


4.5833 


39 


30.6944 


9 


.2600 


39 


46944 


10 


5.1851 


40 


31.8618 


10 


.3086 


40 


49383 


11 


5.8055 


41 


33.0277 


11 


.3734 


41 


5.1883 


12 


&4444 


42 


342222 


12 


.4444 


42 


5.4444 


13 


7.1018 


43 


35.4351 


13 


.6216 


43 


5.7076 


14 


7.7777 


44 


36.6666 


14 


.6049 


44 


5.9753 


15 


8.4720 


45 


37.9166 


15 


.6944 


46 


6.2500 


16 


9.1851 


46 


39.1851 


16 


.7901 


46 


6.5308 


17 


a9166 


47 


40.4720 


17 


.8920 


47 


6.8179 


18 


10.6666 


48 


41.7777 


18 


1.0000 


48 


7.1111 


19 


11.4351 


49 


43.1018 


19 


1.1142 


49 


7.4104 


20 


12.2222 


60 


44.4444 


20 


1.2346 


60 


7.7160 


21 


13.0278 


51 


46.8055 


21 


1.3611 


51 


8.0277 


22 


13.8518 


52 


47.1851 


22 


1.4938 


52 


8.3456 


23 


14.6944 


53 


48.6833 


23 


1.6327 


53 


8.6697 


24 


15.5555 


54 


50.0000 


24 


1.7778 


64 


9.0000 


25 


16.4351 


55 


61.4351 


26 


1.9295 


66 


9.3364 


26 


17.3333 


56 


62.8888 


26 


2.0864 


66 


9.6790 


27 


18.2500 


57 


54.3611 


27 


2.2500 


67 10.0277 


28 


19.1851 


58 


55.8518 


28 


2.4197 


58 ;10.3827 


29 


20.1388 


59 


57.3611 


29 


2.5956 69 10.7438 


30 


21.1111 


60 


68.8688 


30 


2.7778 60 ill.llll 





( xxxiiL ) 




BASE 23— SLOPE i to 1. 




Add. 


i 


Add. 


ssas 


Deduct. 




Dednct. 


as 




9B 




O 




a 




1 


.4397 


31 


26.6609 


1 


.0046 


91 


4.4490 


2 


.9073 


32 


27.8518 


2 


.0185 


32 


4.7407 


Z 


1.4028 


33 


29.1805 


3 


.0416 


33 


6.0416 


4 


1.9259 


34 


30.5369 


4 


.0740 


34 


6.3618 


5 


2.4767 


36 


31.9213 


6 


-1167 


36 


6.6712 


6 


3.0555 


36 


33.8333 


6 


.1667 


36 


6.0000 


7 


3.6619 


37 


34.7731 


7 


.2268 


37 


6.3379 


8 


4.2962 


38 


36.2407 


8 


.2963 


38 


6.6861 


9 


4.9583 


39 


37.7361 


9 


.3760 


39 


7.0416 


10 


5.6480 


40 


39.2592 


10 


.4630 


40 


7.4074 


11 


6.3657 


41 


40.8101 


11 


.5602 


41 


7.7824 


12 


7.1111 


^ 


42.3888 


12 


.6667 


42 


8.1666 


13 


7.8842 


43 


43.9963 


13 


.7824 


43 


8.6612 


14 


8.6851 


44 


46.6296 


14 


.«)74 


44 


8.9^29 


16 


9.6137 


46 


47.2916 


16 


1.0417 


46 


9.3760 


16 


10.3704 


46 


48.9816 


16 


1.1862 


46 


9.7962 


17 


11.2646 


47 


60.6990 


17 


1.3379 


47 


10.2268 


18 


12.1666 


48 


32.4444 


18 


1,5000 


48 


10.6666 


19 


iai064 


49 


64.2176 


19 


1.6713 


49 


11.1167 


20 


14.0740 


6§ 


66.0186 


20 


1.8518 


60 


11.6740 


21 


16.0695 


61 


57.8^2 


21 


2.0417 


61 


12.0416 


22 


16.0925 


62 


69.7037 


22 


2.2407 


62 


12.6186 


23 


17.1435 


63 


6L5880 


23 


2.4491 


63 


13.0046 


24 


18.2222 


64 


63.6000 


24 


2.6667 


54 


13.5000 


25 


19.3286 


65 


66.4397 


25 


2.8935 


65 


14.0046 


26 20.4629 


66 


67.4074 


26 


3.1296 


56 


14.6186 


27 


21.6250 


67 


6a4028 


27 


3.3750 


57 


16.0416 


28 


22.8147 


58 


71.4289 


28 


3.6296 


68 


15.6740 


29 


24.0323 


69 


73.4767 


29 


3.8936 


69 


1&1157 


30 


26.2777 


60 


75.6666 


30 


4.1667 


60 1&6666 

F 





( xxxiv. ) 




BASE 23— SLOPE 1 to 1. 


1 


Add. 


} 

31 


Add. 


Dif. of 
Hts. 


Deduct. 


Dif. of 

Ht8. 


Deduct. 


1 


.4444 


31.0000 


1 


.0062 


31 


5.9321 


2 


.9268 


32 


32.6926 


2 


.0247 


32 


6.3210 


3 


1.4444 


33 


34.2222 


3 


.0666 


33 


6.7222 


4 


2.0000 


34 


35.8888 


4 


.0988 


34 


7.1358 


5 


2.6926 


35 


37.6026 


5 


.1543 


36 


7.6617 


6 


3.2222 


36 


39.3333 


6 


.2222 


36 


8.0000 


7 


3.8888 


37 


41.1111 


7 


.3026 


37 


8.4506 


8 


4.5926 


38 


42.9260 


8 


.3951 


38 


8.9135 


9 


5.3333 


39 


44.7777 


9 


.5000 


39 


9.3888 


10 


6.1111 


40 


46.6666 


10 


.6173 


40 


9.8765 


11 


&9258 


41 


48.6926 


11 


.7469 


41 


10.3766 


12 


7.7777 


42 


50.6665 


12 


.8889 


42 


10.8888 


13 


8.6666 


43 


62.6555 


13 


1.0432 


43 


11.4136 


14 


9.6926 


44 


64.5926 


14 


1.2099 


44 


11.9506 


15 


10.6565 


45 


56.6666 


15 


1.3889 


45 


12.5000 


16 


11.6656 


46 


68.7777 


16 


1.5802 


46 


13.0617 


17 


12.5926 


47 


60.9260 


17 


1.7839 


47 


13.6358 


18 


13.6666 


48 


63.1111 


18 


2.0000 


48 


142222 


19 


14.7777 


49 


65.3333 


19 


2.2284 


49 


14.8209 


20 


16.9268 


60 


67.5926 


20 


2.4691 


50 


15.4321 


21 


17.1111 


51 


69.8888 


21 


2.7222 


51 


16.0555 


22 


18.3333 


52 


72.2222 


22 


2.9876 


52 


16.6913 


23 


19.5926 


63 


74.5926 


23 


3.2664 


53 


17.3395 


24 


20,8888 


64 


77.0000 


24 


3.5555 


64 


18.0000 


26 


22.2222 


66 


79.4444 


25 


3.8530 


66 


18.6728 


26 


23.5926 


66 


81.9258 


26 


4.1728 


56 


19.3680 


27 


25.0000 


57 


84.4444 


27 


4.6000 


57 


20.0565 


28 


26.4444 


58 


87.0000 


28 


4.8396 


68 


20.7654 


29 


27.9268 


69 


89.6926 


29 


6.1913 


59 


21.4876 


30 


29.4444 60 1 


92.2222 30 1 


5.5566 60 


22.2222 



1 




( XXXV. ) 




BASE 23— SLOPE IJ to 1. 


•8 


Add. 


1 


Add. 


if. of 


Deduct. 


if. of 
[Its. 


Deduct. 


a 
1 




K 




P 




31 




.4490 


31 


35.4490 


1 


.0077 


7.4161 


2 


.9444 


32 


37.3333 


2 


.0309 


32 


7.9012 


8 


1.4861 


33 


39.2640 


3 


.0694 


33 


8.4027 


4 


2.0741 


34 


41.2406 


4 


.1234 


34 


8.9197 


5 


2.7081 


36 


43.2640 


6 


.1928 


36 


9.4621 


6 


3.8888 


36 


45.3333 


6 


.2778 


36 


10.0000 


7 


4.1156 


37 


47.4490 


7 


.3781 


37 


10.6632 


8 


4.8888 


38 


49.6111 


8 


.4938 


38 


11.1419 


9 


5.7083 


39 


51.8193 


9 


.6260 


39 


11.7361 


10 


6.5740 


40 


54.0740 


10 


.7716 


40 


12.3456 


11 


7.4862 


41 


66.3760 


11 


.9336 


41 


12.9706 


12 


8.4444 


42 


58.7222 


12 


1.1111 


^ 


13.6111 


13 


9.4490 


43 


61.1157 


13 


1.3040 


43 


14.2680 


14 


10.6000 


44 


63.5566 


14 


1.6123 


44 


14.9382 


Id 


11.5971 


46 


66.0417 


15 


1.7361 


46 


16.6250 


16 


12.7407 


46 


68.5740 


16 


1.9753 


46 


16.3271 


17 


13.9303 


47 


71.1528 


17 


2.2299 


47 


17.0447 


18 


16.1666 


48 


73.7777 


18 


2.6000 


48 


17.7777 


19 


16.4490 


49 


76.4490 


19 


2.7865 


49 


18.5262 


20 


17.7777 


60 


79.1666 


20 


3.0864 


50 


19.2901 


21 


19.1528 


51 


81.9303 


21 


3.4028 


51 


20.0694 


22 


20.5740 


62 


84.7407 


22 


3.7346 


62 


20.8641 


23 


22.0417 


53 


87.5972 


23 


4.0818 


53 


21.6743 


24 


23.5556 


64 


90.5000 


24 


4.4444 


64 


22.5000 


25 


25.1167 


55 


93.4490 


26 


4.8228 


65 


23.3410 


26 


26.7222 


66 


96.4444 


26 


6.2160 


66 


24.1975 


27 


28.3750 


57 


99.4861 


27 


5.6250 


57 


26.0694 


28 


30.0740 


68 


102.5740 


28 


6.0494 


58 


25.9567 


29 


31.8193 


69 


106.7083 


29 


6.4891 


69 


26.8696 


30 


33.6111 


60 


108.8888 


30 


6.9444 


60 


27.7777 

> 



( xxxvi. ) 
BA8E 23— SLOPE IJ to 1. 



1 

•8 


Add. 


1 


Add. 


if. of 


Deduct. 


2-a 


Deduct. 


a 




a 




o 




o 




1 


.4536 


31 


39.8982 


1 


.0092 


31 


8.8981 


2 


.962» 


32 


^0741 


2 


.0370 


32 


9.4815 


3 


1.5278 


33 


44.3055 


3 


.0833 


33 


10.0838 


4 


2.1481 


34 


4&6026 


4 


.1480 


34 


10.7037 


5 


2.8238 


35 


48.9352 


6 


.2313 


36 


11.3425 


6 


3.5555 


36 


51.3333 


6 


.3333 


36 


12.0000 


7 


4.3426 


37 


6a78e9 


7 


.4537 


37 


12.6759 


8 


6.1861 


38 


66.2964 


8 


.5926 


38 


13.370a 


9 


6.0833 


39 


68.8610 


9 


.7500 


89 


14.063» 


10 


7.0370 


40 


31.4814 


10 


.926» 


40 


148148 


11 


8.0464 


41 


641574 


11 


1.1203 


41 


16.6648 


12 


9.1111 


42 


66.8888 


12 


1.3333 


42 


16.3333 


13 


10.2314 


431 


69.6759 


13 


1.6648 


43 


17.1202 


14 


11.4074 


44 


72.6185 


14 


1.8148 


44 


17.9259 


15 


12.6387 


45 


76.4167 


15 


2.0833 


45 


18.7600 


16 


13.9259 


46 


78.3703 


16 


2.3704 


46 


19.6926 


17 


16.2682 


47 


81.3796 


17 


2.6769 


47 


20.4537 


18 


16.6666 


48 


84.4444 


18 


3.0000 


48 


21.3333 


19 


18.1203 


^ 


87.6647 


19 


8.3426 


49 


22.2314 


20 


19.6296 


50 


90.7407 


20 


3.7037 


60 


2ai481 


21 


21.1945 


61 


93.9720 


21 


40833 


61 


24.0833 


22 


22.8148 


52 


97.2892 


22 


44816 


52 


25.0370 


23 


24.^68 


63 


100.6018 


2a 


48981 


53 


26.0092 


24 


26.2222 


54 


1040000 


24 


6.3333 


64 


27.0000 


25 


28.0092 


55 


107.4536 


26 


6.7860 


66 


28.0092 


26 


29.8618 


50 


110.9^29 


26 


6.2592 


66 


29.0370 


27 


31.7600 


57 


1145278 


27 


6.7500 


57 


80.0833 


28 


33.7087 


58 


118.1481 


28 


7.2692 


68 


31.1481 


29 


36.7128 


69 


121.8238 


29 


7.7869 


59 


32.2314 


30 


37.7777 


60 


126.6566 


30 


8.3333 


60 


33.3383 



( xzxvii. ) 
BASE 23— SLOPE 1» to 1. 



1 


Xdi. 


1 


Add. 


if. of 


Deduct. 


if. of 


Deduct. 


s 


• 


n 




1 




Q 




1 


.4581 


31 


443473 


.0108 


31 


10.3811 


2 


.9815 


32 


46.8148 


2 


.0432 


32 


11.0609 


3 


1.5695 


33 


'^.3473 


3 


.0972 


33 


11.7638 


4 


2.2222 


34 


51.9444 


4 


.1728 


34 


12.4876 


5 


2.9395 


35 


546065 


5 


.2700 


35 


13.2330 


& 


3.7222 


36 


57.3333 


6 


.3889 


36 


14.0000 


7 


4.5696 


37 


60.1250 


7 


.5293 


37 


147885 


8 


5.4814 


38 


62.9815 


8 


.6913 


38 


15.6987 


9 


6.458a 


39 


65.9027 


9 


.8760 


39 


1&4305 


10 


7.6000 


40 


68.8888 


10 


1.0602 


40 


17.2839 


11 


8.6065 


41 


71.9399 


11 


1,3071 


41 


18.1589 


12 


9.7777 


42 


75.0555 


12 


1.5555 


42 


190565 


13 


11.0138 


43 


78,2362 


13 


1.8256 


43 


19.9747 


14 


12.3147 


44 


81.4814 


14 


2.1173 


44 


20.9135 


15 


ia6803 


45 


84.7916 


15 


2.4305 


45 


21.8750 


16 


15.1111 


46 


88.1666 


16 


2.7654 


46 


22.8680 


17 


16.6065 


47 


91.6065 


17 


3.1219 


47 


23.8626 


18 


18.1666 


48 


95.1111 


18 


3.5000 


48 


24.8888 


19 


19.7916 


49 


98.6803 


19 


3.8997 


49 


26.9367 


20 


21.4814 


50 


102.3147 


20 


43210 


50 


27.0061 


21 


23.2362 


51 


106.0138 


21 


47639 


51 


28.0972 


22 


25.0555 


52 


109.7777 


22 


6.2284 


52 


29.209& 


23 


26.9399 


53 


113.6065 


23 


5.7145 


53 


30.3441 


24 


28.8888 


54 


117.6000 


24 


6.2222 


54 


31.5000 


25 


30.9027 


55 


121.4583 


25 


6.7516 


55 


32.6774 


26 


32.9815 


56 


125.4814 


26 


7.3024 


56 


33.8765 


27 


35.1250 


57 


129.5696 


27 


7.8750 


57 


35.0972 


28 


37.3333 


58 


133.7222 


28 


8.4691 


68 


36.3394 


29 


39.6065 


59 


137.9395 


29 


9.0648 


69 


37.6033 


30 


41.9444 


60 


142.2222 


30 


9.7222 


60 


tJOaOOOO 





( xxxviii. ) 






BASE 23— SLOPE 2 to 1. 

• 




M Hdght 


Add. 


•4^ 

•a 

•s 


Add. 


tax 

a 
1 


Deduct. 


a 

31 


Dedact. 


.4629 


31 


48.7962 


.0123 


11.8642 


2 


1.0000 


32 


51.5555 


2 


.0494 


32 


12.6419 


3 


1.6111 


33 


54.3888 


3 


.1111 


33 


13.4444 


4 


2.2962 


34 


57.2962 


4 


.1975 


34 


14.2716 


5 


3.0555 


35 


60.2777 


5 


.3086 


35 


16.1234 


6 


3.8888 


36 


63.3333 


6 


.4444 


36 


16.0000 


7 


4.7962 


37 


66.4629 


7 


.6049 


37 


16.9012 


8 


5.7777 


38 


69.6666 


8 


.7901 


38 


17.8271 


9 


6.8333 


39 


72.9444 


9 


1.0000 


39 


18.7777 


10 


7.9629 


40 


76.2962 


10 


1.2346 


40 


19.7530 


11 


9.1666 


41 


79.7222 


11 


1.4938 


41 


20.7530 


12 


10.4444 


42 


83.2222 


12 


1.7778 


42 


21.7777 


13 


11.7962 


43 


86.7962 


13 


2.0864 


43 


22.8271 


14 


13.2222 


44 


90.4444 


14 


2.4197 


44 


23.9012 


15 


14.7222 


45 


94.1666 


15 


2.7778 


46 


25.0000 


16 


16.2962 


46 


97.9689 


16 


3.1605 


46 


26.1234 


17 


17.9444 


47 


101.8333 


17 


3.5679 


47 


27.2716 


18 


19.6666 


48 


105.7777 


18 


4.0000 


48 


28.4444 


19 


21.4629 


49 


109.7962 


19 


4.4568 


49- 


29.6420 


20 


23.3333 


50 


113.8888 


20 


4.9382 


50 


30.86^ 


21 


25.2777 


51 


118.0555 


21 


5.4444 


51 


32.1111 


22 


272962 


52 


122.2962 


22 


6.9753 


52 


33.3827 


23 


29.3888 


53 


126.6111 


23 


6.5309 


63 


34.6790 


24 


31.5555 


54 


131.0000 


24 


7.1111 


54 


36.0000 


25 


33.7962 


55 


135,4629 


25 


7.7160 


65 


37.3456 


26 


36.1111 


56 


140.0000 


26 


8.3457 


56 


38.7160 


27 


38.5000 


67 


144.6111 


27 


9.0000 


57 


40.1111 


28 


40.9629 


58 


149.2962 


28 


9.6790 


58 


41.5308 


29 


43.5000 


59 


154.0555 


29 


10.3827 


59 


42.9753 


30 


46.1111 


60 


158.8888 1 30 


11.1111 


60 


44.4444 







( xzxix. ) 




BASE 23— SLOPE 2J to 1. 


•s 


Add. 


S 


Add. 


5a BB 


Deduct. 




Deduct. 


a 




X 




Q 




Q 




1 


.4722 


31 


67.6946 


1 


.0154 


31 


148302 


2 


1.0370 


32 


61.0370 


2 


.0617 


32 


16.8013 


3 


1.6946 


33 


644722 


3 


.1389 


33 


16.8055 


4 


2.4444 


34 


68.0000 


4 


.2468 


34 


17.8396 


5 


3.2870 


36 


71.6203 


6 


.3857 


35 


18.9043 


6 


42222 


36 


75.3333 


6 


.5566 


36 


20.0000 


7 


6.2500 


37 


79.1387 


7 


.7661 


37 


21.1266 


8 


6.3703 


38 


83.0370 


8 


.9876 


38 


22.2839 


9 


7.6833 


39 


87.0276 


9 


1.2600 


39 


23.4722 


10 


8.8888 


40 


91.1111 


10 


1.6432 


40 


24.6913 


11 


10.2870 


41 


96.2870 


11 


1.8673 


41 


25.9413 


12 


11.7777 


42 


99.5655 


12 


2.2222 


42 


27.2222 


13 


13.3611 


43 


103.9166 


13 


2.6080 


43 


28.5360 


14 


16.0370 


44 


108.3703 


14 


3.0247 


44 


29.8765 


16 


16.8056 


46 


112.9166 


16 


3.4722 


45 


31.2600 


16 


18.6666 


46 


117.6665 


16 


3.9606 


46 


32.6643 


17 


20.6203 


47 


122.2870 


17 


44599 


47 


34.0895 


18 


22.6666 


48 


127.1111 


18 


5.0000 


48 


35.5556 


19 


24.8056 


49 


132.0277 


19 


5.5710 


49 


37.0524 


20 


27.0370 


60 


137.0370 


20 


6.1728 


50 


38.5802 


21 


29.3611 


61 


142.1388 


21 


6.8055 


61 


40.1388 


22 


31.7777 


52 


147.3333 


22 


7.4691 


62 


41.7283 


23 


34.2870 


63 


152.6203 


23 


8.1636 


63 


43.3487 


24 


36.8888 


64 


158.0000 


24 


8.8889 


54 


45.0000 


26 


39.5833 


65 


163.4722 


25 


9.6466 


66 


46.6820 


26 


42.3703 


66 


169.0370 


26 


10.4321 


56 


48.3950 


27 


46.2600 


57 


1746946 


27 


11.2600 


67 


60.1388 


28 


48.2222 


58 


180.4444 


28 


12.0587 


58 


51.9136 


29 


61.2870 


69 


186.2870 


29 


12.9782 


69 


63.7191 


30 


644444 


60 


192.2222 


30 


13.8889 


60 


66.6556 







(xl.) 




BASE 23— SLOPE 3 to 1. 


1 


Add. 


1 


Add. 




Dedact. 


2^ 


Deduct. 


X 




X 




1 




« 




1 


.4814 


31 


66.5926 


.0185 


31 


17.7963 


2 


1.0740 


32 


70.5185 


2 


.0740 


32 


18.9630 


3 


1,7777 


33 


745555 


3 


.1667 


33 


20.1666 


4 


2.5926 


34 


78.7037 


4 


.2961 


34 


21.4074 


5 


3.5185 


35 


82.9629 


5 


.4628 


35 


22.6861 


6 


45555 


36 


87.3333 


6 


.6667 


36 


24.0000 


7 


5.7037 


37 


91.8147 


7 


.9074 


37 


25.3518 


8 


&g629 


38 


96.4074 


8 


1.1852 


38 


26.7407 


9 


8.3333 


39 


101.1111 


9 


1.6000 


39 


28.1666 


10 


9.8147 


40 


105.9259 


10 


1.8518 


40 


29.6296 


11 


11.4074 


41 


110.8518 


11 


2.2407 


41 


31.1296 


12 


13.1111 


42 


115.8888 


12 


2.6667 


42 


32.6666 


13 


149-259 


43 


121.0370 


13 


3.1296 


43 


34.2405 


14 


16.8518 


44 


126.2962 


14 


3.6296 


44 


36.8518 


15 


18.8888 


45 


131.6666 


15 


41667 


46 


37.5000 


16 


21.0370 


46 


137.1481 


16 


47407 


46 


39.1851 


17 


23.2962 


47 


142.7407 


17 


5.3518 


47 


40.9074 


18 


25.6666 


48 


148.4444 


18 


6.0000 


48 


42.6666 


19 


28.1481 


^ 


154.2592 


19 


a6852 


49 


44.4629 


20 


30.7407 


50 


160.1852 


20 


7.4074 


50 


46.2963 


21 


33.4444 


51 


16a2222 


21 


8.1667 


61 


49.1666 


fl2 


3&2592 


52 


172.3703 


22 


8.9629 


62 


50.0740 


23 


39.1852 


53 


178.6296 


23 


9.79^ 


63 


52.0185 


24 


42.2022 


54 


185.0000 


24 


10.6667 


54 


54.0000 


25 


45.3703 


55 


191.4814 


25 


11.6741 


55 


56.0184 


26 


48.6296 


56 


198.0740 


26 


12.5184 


56 


68.0640 


27 


52.0000 


57 


204.7777 


27 


13.5000 


57 


60.1666 


28 


55.4814 


58 


211.5926 


28 


145185 


58 


62.2962 


29 


69.0740 


69 


218.6186 


29 


16.6739 


69 


64.4629 


30 


62.7777 


60 


225.5555 


SO 


16.6667 


60 


66.6666 



(xli.) 
BASE 24— SLOPE i to 1. 



-a, 

•s 



1 

2 
3 
4 
5 

6 
7 

8 

9 

10 

11 
12 
13 
14 
15 

16 
17 
18 
19 
20 

21 
22 
23 
24 
25 

26 
27 
28 
29 
30 



Add. 



.4490 

.9074 

1.3750 

1.8518 

2.3379 

2.8333 
3.3379 

3.8518 
4.3750 
4.9074 

5.4490 
6.0000 
6.5601 
7.1-296 
7.7083 

8.2962 

8.8935 

9.5000 

10.1157 

10.7407 

11.3750 
12 0185 
12.6712 
13.3333 
14.0046 

14.6851 
15.3750 
16.0740 
16.7824 
17.5000 






Add. 



31 
32 
33 
34 
35 

36 
37 
38 
39 
40 

41 
42 
43 
44 

45 

46 
47 
48 
49 
50 

51 
52 
53 
54 
55 

56 
57 
58 
59 
60 



18.2268 
18.9629 
19.7083 
20.4629 
21.2268 

22.0000 
22.7824 
23.5740 
24.3750 
25.1851 

26.0046 
26.8333 
27.6712 
28.5185 
29.3750 

30.2407 
31.1157 
32.0000 
32.8935 
33.7962 

347083 
35.6296 
36.5601 
37.5000 
38.4490 

39.4074 
40.3750 
41.3518 
42.3379 
43.3333 






1 

2 
3 
4 
5 

6 
7 

8 

9 

10 

11 
12 
13 
14 
15 

16 
17 
18 
19 
20 

21 
22 
23 
24 
25 

26 
27 
28 
29 
30 



Deduct. 



.0015 
.0062 
.0139 
.0246 
.0385 

.0555 
.0756 
.0988 
.1250 
.1543 

.1867 
.2222 
.2608 
.3025 
.3472 

.3951 
.4460 
.5000 
.5571 
.6173 

.6805 
.7469 
.8163 
.8889 
.9647 

1.0432 
1.1250 
1.2099 
1.2978 
1.3889 






31 
32 
33 
34 
35 

36 
37 
38 
39 
40 

41 
42 
43 
44 
45 

46 

47 
48 
49 
50 

51 
52 
53 
54 
55 

56 
57 
58 
59 
60 
a 



Deduct. 

1.4830 
1.5802 
1.6805 
1.7839 
1.8904 

2.0000 
2.1126 
2.2284 
2.3472 
2.4691 

2.5941 
2.7222 
2.8545 
2.9876 
3.1250 

3.2654 
3.4089 
3.5555 
3.7052 
3.8580 

4.0139 
4.1728 
4.3349 
4.5000 
4.6682 

4.8395 
5.0139 
5.1913 
5.3719 
5.5555 



I 

\ 



( xlii. ) 


BASE 24^8LOPE J to 1. 


•s 


Add. 


•s 


Add. 


S3 35 


Deduct. 




Deduct. 


n 
1 


- 


s 




o" 




31 




. .4536 


31 


22.6759 


1 


.0031 


2.9660 


-.2 


.9259 


32 


23.7037 


2 


.0123 


32 


3.1605 


3. 


.1.4167 33 
.1.9269 34 


24.7600 


3 


.0278 


33 


36111 


4. 


25.8147 


4 


.0494 


34 


3.5679 


5 


2.4636 


35 


26.8981 


5 


.0772 


36 


3.7808 


6 


3.0000 


36 


28.0000 


6 


,1111 


36 


4.0000 


7 


3.5647 


37 


29.1203 


7 


.1512 


37 


4.2253 


8 


4.1481 


38 


30.2692 


8 


.1975 


38 4.4668 


9 


4.7600 


39 31.4166 


9 


.2600 


39 


4.6944 


10 


6.3703 


40 32.6926 


10 


.3086 


40 


4.9383 


11 


6.0092 


41 


33.7870 


11 .3734 


41 


5.1883 


12 


6.6666 


^ 


36.0000 


12 


AAAA 

• X X J. X 


^ 


5.4444 


13 


7.3^25 


43 


36.2314 


13 


.5216 


43 


6.7067 


14 


8.0370 


44 


37.4814 


14 


.6049 


44 


5.9763 


16 


8.7500 


45 


38.7600 


15 


.6944 


46 


6.2500 


16 


9.4814 


46 


40.0370 


16 


.7901 


46 


6.5308 


17 


10.2314 


47 


41.3426 


17 


.8920 


47 


6.8179 


18 


11.0000 


48 


'^.6666 


18 


1.0000 


48 


7.1111 


19 


11.7870' 


49 


44.0092 


19 


1.1142 


49 


7.4104 


20 


12.6926 


60 


46.3703 


20 


1.2346 


60 


7.7160 


21 


13.4167 


61 


46.7500 


21 


1.3611 


51 


8.0277 


22 


14.2592 


62 


48.1481 


22 


1.4938 


62 


8.3466 


23 


16.1203 


53 


49.5647 


23 


1.6327 


63 


8.6697 


24 


16.0000 


54 


51.0000 


24 


1.7778 


64 


9.0000 


25 


1&8981 


66 


62.4536 


25 


1.9295 


55 


9.3364 


26 


17.8147 


56 


63.9269 


26 


2.0864 


56 


9.6790 


27 


18.7500 


57 


66.4167 


27 


2.2500 


67 


10.0277 


28 


19.7037 


58 


66.9269 


28 


2.4197 


58 


1038-i7 


29 


20.6759 


69 


58.4536 


29 


2.5956 


5;) 


10.7438 


30 


21.6666 


60 


60.0000 


30 


2.7778 


60 


11.1111 



( xliii. ) 


BASE- 24— SLOPE f to 1. 


■a 

•s 


Add. 


•t 


Add. 




Deduct.' 




Deduct. 


1 


- • •TtvO^, 


31 


27.1250 


1 


.0046 


31 


4.4490 


2 


.•••ty^^TC 


32 


28.4444 . 


2 


.0185 


32 


'4.7407 


3 


a*:458,4; 


33 


29.7916 


^ o« 


<.041«; 


33 


6.^16 


4 


2:0600. 


34 


31.1666 


•■'4 


.0740. 


^ 


5.3618 


5 


2.5692; 


35 

1 


; 32.5692 


1/ 6 


11157 


35 


66712 


6 


3.1666 


36. 


S* 34.0000 


6 


.1667^ 


36 


6.0000 


7 


3.7915 


37.' 


/.-35.4582 


"7 


.2268" 


37 


6.3379 


8 


4.4444 


38 


n 36.9444 


8 


.296S 


38 


6.6851 


9 


5.1250 


39 


• 38.4583 


9 


.3750 


39 


7.0416 


10 


5.8333 


40 


40.0000 


10 


.4630 


40 


7.4074 


11 


6.5694 


41 


41.5692 


11 


.5602 


41 


7.7824 


12 


7.3333 


42 


43.1666 


12. 


.6667 


42 


8.1666 


13 


8.1250 


43 


44.7916 


13 


-.7824 


43 


'8:6612 


14 


8.9444 


44 


46.4444 


14. 


.9074 


44 


8.9629 


15. 


9.7915 


45 


48.1250 


16 


1.0417 


45 


9.3750 


tec 


KB6666 


46 


49.8333 


16 


1.1852 


46 


'9:7962 


17jt 


iai56g2 


47 


51.5692 


17. 


1».3379 


47 


10V2268 


I8& 


1-2^5000 


48 


53.3333 


18. 


115000 


48 


10.6666 


m 


13.4584 


49 


55.1250 


19 


1.6713 


49 


11.1157 


20. 


14.4444 


60 


56.9444 


20 


1.8518 


50 


11.5740 


21 


15.4584 


51 


: 58.7916 


21 


20417 


51 


12.0416 


22t 


16.5000, 


52 


.60.6666 


22 


2.2407 


62 


12.6186 


• ^O V 


t:?:5692 


•63i 


;.62.5692 


23 


2.4491 


63 


13.0046 


24, 


*8)666a 


,54 


•64.5000 


24 


2.6667 


64 


13.5000 


25.1 


19.7916. 


55 


' 66.'4584 


25 


2.8935 


56 


14.0046 


26- 


20:9444 


56- 


68.4444 


26 


3.1296 


56 


14.5185 


27, 


22il250 


67 


70.4584 


27 


3.3750 


57 


15.0416 


28 


23.3333 


68 


72.5000 


28 


3.6296 


58 


15.6740 


29 


24.5692 


59 


74.5693 


29 


3.8935 


59 


16.1157 


30 


25.8333 


60 


76.6666 


30 


4.1667 


60 


16.6666 







( ^liv. ) 






BASE 24— SLOPE 1 to 1. 




1 


Add. 


4J 


Add. 


if. of 
its. 


Deduct. 




Dedaot. 


iS 




n 




Q" 




Q 




1 


,4628 


31 


31.5741 


I 


.0062 


31 


5.9321 


2 


,9629 


32 


33.1851 


2 


.0247 


32 


6.3210 


3 


1.6000 


33 


34.3333 


3 


.0565 


33 


6.7222 


4 


2.0740 


34 


36.5185 


4 


.0988 


34 


7.1368 


5 


2.6850 


35 


38.2407 


5 


.1543 


35 


7.6617 


6 


3.3333 


36 


40.0000 


6 


.2222 


36 


8.0000 


7 


4.0185 


37 


41.7961 


7 


.3026 


37 


8.4606 


8 


4.7407 


38 


43.6296 


8 


.395 L 


38 


8.9135 


9 


5.5000 


39 


45.5000 


9 


.6000 


39 


9.3888 


10 


6.2962 


40 


47.4074 


10 


.6173 


40 


9.8765 


11 


7.1296 


41 


49.3518 


11 


.7469 


41 


10.3765 


12 


8.0000 


42 


51.3333 


12 


.8889 


42 


10.8888 


13 


8.9074 


43. 


63.3518 


13 


1.0432 


43 


11.4136 


14 


9.8518 


44 


65.4073 


14 


1.2099 


44 


11.9506 


15 


10.8333 


45 


57.6000 


15 


1.3889 


45 


12.6000 


16 


11.8518 


46 


59.6296 


16 


1.5802 


46 


13.0617 


17 


12.9074 


47 


61.7962 


17 


1.7839 


47 


13.6368 


18 


14.0000 


48 


64.0000 


18 


2.0000 


48 


14.2222 


19 


16.1296 


49 


66.2407 


19 


2.2284 


49 


14.8209 


2D 


16.2962 


50 


68.5185 


20 


2.4691 


50 


15.4321 


21 


17.5000 


51 


70.3333 


21 


2.7222 


51 


16.0655 


22 


18.7407 


52 


73.1851 


22 


2.9876 


52 


16.6913 


23 


20.0185 


63 


75.6741 


23 


3.2664 


53 


17.3395 


24 


21.3333 


54 


78.0000 


24 


3.5556 


64 


18.0009 


25 


22.6860 


55 


80.4628 


26 


3.8630 


56 


18.6728 


26 


24.0740 


56 


82.9629 


26 


4.1728 


56 


19.3680 


27 


26.5000 


67 


86.6000 


27 


4.6000 


57 


20.0565 


28 


26.9629 


68 


88.0741 


28 


4.8395 


68 


20.7654 


29 


28.4629 


69 


90.6860 


29 


5.1913 


59 


21.4876 


30 


30.0000 


60 


93.3333 


30 


5.5665 


60 


22.2322 







( »lv. ) 








BASE 24— SLOPE 1^ to 1, 




t 


Add. 


4^ 

1 


Add. 




|"S . 

Deduct. I:«3m 


Deduct. 


X 




X 




q" 




O" 




1 


.4674 


31 


36.0232 


1 


jomi 


31 


7.4151 


2 


.9814 


32 


37.9269 


2 


.0309 


32 


7.9012 


8 


1.6418 


33 


89.8760 


3 


.0694 


33 


8.4027 


4 


2.1482 


34 


41.8703 


4 


.1284 


34 


8.9197 


5 


2.8007 


36 


43.9120 


6 


.1928 


36 


9.4521 


6 


3.6000 


36 


46.0000 


6 


.2778 


36 


10.0000 


7 


4.2463 


37 


48.1340 


7 


.8781 


37 


10.5632 


8 


6.0370 


38 


60.3148 


8 


.4938 


38 


11.1419 


9 


5.8750 


39 


62.6416 


9 


.6250 


39 


11.7361 


10 


&7692 


40 


64.8148 


10 


.7716 


40 


12.3456 


11 


7.6898 


41 


67.1342 


11 


.9336 


41 


12.9706 


12 


8.6666 


42 


69.5000 


12 


1.1111 


42 


13.6111 


13 


9.6898 


43 


61.9120 


13 


1.8040 


48 


14.2680 


14 


10.7592 


44 


64.3703 


14 


1.5123 


44 


14.9382 


15 


11.8760 


46 


66.8750 


16 


1.7361 


46 


15.6250 


16 


13.0370 


46 


69.4259 


16 


1.9763 


46 


16.3271 


17 


14.2463 


47 


72.0231 


17 


2.2299 


47 


17.0447 


18 


16.5000 


48 


74.6666 


18 


2.5000 


48 


17.7777 


19 


16.8007 


49 


77.3564 


19 


2.7856 


49 


18.5262 


20 


18.1481 


50 


80J0926 


20 


3.0864 


60 


19.2901 


21 


19.6418 


51 


82.8750 


21 


3.4028 


51 


20.0694 


22 


20.9814 


52 


85.7086 


22 


3.7846 


52 


20.8641 


23 


22.4674 


53 


88.5785 


23 


4.0818 


63 


21.6743 


24 


24.0000 


54 


91.6000 


24 


4.4444 


54 


22.5000 


26 


26.6786 


66 


94.4674 


26 


4.8228 


55 


23.3410 


26 


27.2036 


56 


97.4814 


26 


6.2160 


56 


24.1975 


27 


28.8750 


67 


100.6418 


27 


6.^50 


57 


25.1694 


28 


30.5925 


68 


103.6481 


28 


6.0494 


68 


25.9567 


29 


32.3564 


69 


106.8007 


29 


6.4891 


69 


26.8595 


30 


34.1666 


60 


110.0000 


30 


6.9444 


60 


27,7777 





( xlvi. ) 




BASE 24— SLOPE IJ to 1. 




•a 


Add. 


4^ 

■a 

•c 


Add. 


O J 


Deduct. 




Deduct. 




93 
1 




s 




1 




31 






.4720 


31 


40.4722 


.0092 


8.8981 




2 


1.0000 


32 


42.6666 


2 


.0370. 


32 


9,4815 




3 


1.5833 


33 


44.9166 


3 


.0833 


33 


10.0833 




4 


2.2222 


34 


47.2222 


4 


.1480 


34 


10.7037 




5 


2.9162 


36 


49.6833 


5 


.2313 


36 


11.3426 




6 


3.6666 


36 


52.0000 


6 


.3333 


36 


12:0000 




7 


4.4722 


37 


64.4719 


7 


.4637 


37. 


12.'6759 




8 


5.3333 


38 


67.0000 


8 


.5926 


38. 


13.3703 




9 


6.2500 


39 


59.5833 


9 


.7500 


39. 


14.0833 




H) 


7.2222 


40 


62.2222 


10 


.9259 


40, 


14.8148 




11 


8.2500 


41 


64.9166 


11 


1.1203 


41. 


15.5648 




! 12 


9.3333 


42 


67.6666 


12 


1.3333 


42 


16.3333 




13 


10.4722 


43 


70.4722 


13 


1.5648. 


43 


17.1202 




14 


11.6666 


44 


73.3333 


14 


1.8148 


44 


17.9269 




16 


12.9162 


45 


76.2600 


15 


2.0833. 


46 


18.7600 




16 


14.2222 


46 


79.2222 


16 


2.3704- 


46 


19.5925 




17 


16.5833 


47 


82.2600 


17. 


2:6759. 


47 


20.4637; 




18 


17.0000 


48 


86.3333 


18 


3.0000. 


48.. 


21:3333. 




19 


18.4720 


49 


88.4720 


19 


3.3426 


49; 


22:2314 




20 


20.0000 


50 


92.6666 


.20 


3.7037 


60 


23.1481 . 




21 


21.5833 


51 


94.9166 


21 


4.0833. 


61 


24,0833 




22 


23.2222 


62 


98.2222 


22 


4.4816 


62.. 


2^.P370 




23 


24.9166 


53 


101.5833 


23 


4.8981 


63. 


26:0092 




24 


26.6666 


54 


106.0000 


24 


6:3333 


64 


27.0000 




25 


28.4720 


66 


108.4720 


25 


5:7869 


55 


28.0092 




26 


30.3333 


66 


112.0000 


26 


6:2592 


66 


29.0370 




27 


32.2500 


67 


115.5833 


27 


6.7500. 


57 


30.0833 




28 


34.2222 58 


119.2222 


28 


7.2592 


58 


31.1481 




29 


36.2500 69 


122.9162 


29 


7.7869, 


69 


32.2314 


1 


30 


38.3333 60 


126.6666 


30 


8.3333 


60 


33.3333 


1 



( xlvii. ) 
BASE 24— SLOPE Ij to 1. 



Height 


• Add. 


t 

•s 
31 


Add. 


Dif. of 


Deduct. 


(4-1 

31 


Deduct. 


1 


.4768 


44.9214 


1 


.0108 


10.3811 


2 


1.0185 


32 


47.4073 


2 


.0432 


32 


11.0609 


3 


1.6250 


33 


49.9583 


3 


.0972 


33 


1 1.7638 


4 


2.2963 


34 


62.5740 


4 


.1728 


34 


12.4876 


5 


3.0319 


35 


56.2646 


6 


.2700 


35 


13.2330 


6 


3.8333 


36 


68.0000 


6 


.3889 


36 


14.0000 


7 


4.6990 


37 


60.8102 


7 


.6293 


37 


14.7885 


8 


5.6296 


38 


63.6862 


8 


.6913 


38 


15.5987 


9 


6.6250 


39 


66.6250 


9 


.8760 


39 


16.4305 


10 


7.6851 


40 


69.6296 


10 


1.0802 


40 


17.2839 


11 


8.8102 


41 


72,6990 


11 


1.3071 


41 


18.1589 


12, 


10.0000 


42 


76.8333 


12 


1.5555 


42 


19.0555 


13 


11.2546 


43 


79.0319 


13 


1.8266 


43 


19.9747 


14 


12:5740 


44 


82.2962 


14 


2.1173 


44 


20.9135 


15 


13.9583 


45 


85.6-250 


15 


2.4306 


45 


21.8750 


16 


15.4074 


46 


89.0186 


16 


2.7654 


46 


22.8580 


17 


16,y214 


47 


92.4768 


17 


3.1219 


47 


23.8626 


18 


18.5000 


48 


96.0000 


18 


3.6000 


48 


24.8888 


19 


20.1435 


49 


99.6878 


19 


3.8997 


49 


25.9367 


20 


21.8518 


50 


104.2407 


,20 


4.3210 


50 


27.0061 


21 


23.6250 


51- 


106.9683 


21 


4.7639 


51 


28.0972 


22 


25.4629 


62 


110.7407 


22 


6.2284 


52 


29.2098 


ii3 


27.3658 


53 


114.6878 


23 


6.7145 


53 


30.3441 


24 


29.3333 


54 


118.51)00 


24 


6.2222 


54 


31.5000 


25 


31.3658 


65. 


122.4768 


25 


6.7516 


55 


32.6774 


26 


33.4629 


56 


126.5186 


26 


7.3024 


56 


33.8765 


27 


35.6250 


67 


l:i0.6'.i50 


27 


7.8750 


57 


35.0972 


28 


37.8518 


58 


134.7962 


28 


8.4691 


58 


36.^394 


29 


40.1435 


69 


139.0319 


29 ,9.0848 


59 


37.6033 


30 


42.5000 


60 


143.3333 


30 1 


9.7222 1 


60 


38.8888 



( xlviii. ) 
BASE 24-.SLOPE 2 to 1. 



•4^ 


Add. 


SB 

31 


Add. 


gSB 
1 


Deduct. 


<»4 

31 


Deduct. 


1 


.4814 


49.3704 


.0123 


11.8642 


2 


1.0370 


32 


52.1480 


2 


.0494 


32 


12.6419 


8 


1.6666 


33 


65.0(100 


3 


.1111 


33 


13.4444 


4 


2.3704 


34 


57.9257 


4 


.1975 


34 


14.2716 


6 


3.1480 


35 


60.9257 


5 


.3086 


35 


15.1234 


6 


4.0000 


36 


64.0000 


6 


.4444 


36 


16.0000 


7 


4.9257 


37 


67.1480 


7 


.6049 


37 


16.9012 


8 


5.9257 


38 


70.3704 


8 


.7901 


38 


17.8271 


9 


7.00(J0 


39 


73.6666 


9 


1.0000 


39 


18.7777 


10 


8.1480 


40 


77.0370 


10 


1.2346 


40 


19.7630 


11 


9.3704 


41 


80.4814 


11 


1.4938 


41 


20.7530 


12 


10.6666 


42 


84.0000 


12 


1.7778 


42 


21.7777 


13 


12.0370 


43 


87.5926 


13 


2.0864 


43 


22.8271 


14 


13.4814 


44 


91.2690 


14 


2.4197 


44 


23.9012 


15 


15.0000 


45 


95.0000 


15 


2.7778 


45 


25.0000 


16 


16.5926 


46 


98.8149 


16 


3.1605 


46 


26.1234 


17 


18.2590 


47 


102.7036 


17 


3.5679 


47 


27.2716 


18 J20.0000 


48 


106.6666 


18 


4.(X)00 


48 


28.4444 


19 21.8150 


49 


110.7036 


19 


4.4568 


49 


29.6420 


20 23.7036 


50 


115.8149 


20 


4.9382 


50 


30.8642 


21 


25.6666 


51 


119.0000 


21 


5.4444 


51 


32.1111 


22 


27.7036 ! 52 


123.2890 


22 


5.9753 


52 


33.3827 


23 129.8149:53 


127.5926 


23 


6.5309 


53 34.6790 


24 j32.0000|54 


132.0000 


24 


7.1111 


64 36.0000 


25 34.-2590J 55 


136.4814 


25 


7.7160 


56 37.3456 


26 36.5926 


56 


141.0370 


26 


8.3457 


66 38.7160 


27 '39.0000:57 


145.0H66 


27 


9.0000 


57 


40.1111 


28 


41.4814(58 


150.3704 


28 


9.6790 


68 


41.5308 


29 


44.0370 : 59 


165.1480 


29 


10.3827 


59 


42.9753 


30 


46.6666 


160 


160.0000] 


30 


11.1111 


60 


44.4444 





( xlix. ) 




BASE 24— SLOPE 2J to 1. 


•s 


Add. 


1 

s 


Add. 


Dif.of 
Hts. 


D«dact. 


Dif.of 

Hts. 


Deduct. 


1 


.4906 


31 


68.2686 


1 


.0164 


31 


148302 


2 


1.0740 


32 


61.6296 


2 


.0617 


32 


15.8016 


3 


1.7600 


33 


66.0834 


3 


.1389 


33 


16.8065 


4 


2.5185 


34 


68.^96 


4 


.2468 


34 


17.8395 


5 


3.3796 


35 


72.2686 


5 


.3867 


35 


18.9043 


6 


43333 


36 


76.0000 


6 


.5555 


36 20.0000 1 


7 


6.3796 


37 


79.8240 


7 


.7661 


37 


21.1266 


8 


6.6185 


38 


83.7407 


8 


.9876 


38 


22.2839 


9 


7.7600 


39 


87.7600 


9 


1.2500 


39 


23.4722 


10 


9.0740 


40 


91.8518 


10 


1.6432 


40 


246913 


11 


10.4900 


41 


96.0460 


11 


1.8673 


41 


25.9413 


12 


12.0000 


^ 


100.3333 


12 


2.2222 


42 


27.2222 


13 


13.6018 


43 


104.7131 


13 


2.6080 


48 


28.5360 


14 


15.2962 


44 


109.1851 


14 


3.0247 


44 


29.8765 


15 


17.0833 


46 


113,7500 


16 


3.4722 


45 


31.2500 


16 


18.9629 


46 


118.4074 


16 


8.9506 


46 


32.6543 


17 


20.93^ 


47 


123.1576 


17 


44699 


47 


340896 


18 


23.0000 


48 


128.0000 


18 


6.0000 


48 


35.5565 


19 


26.1576 


49 


132.9349 


19 


5.6710 


49 


37.0624 


20 


27.4074 


50 


138.9629 


20 


6.1728 


50 


38.5802 


51 


2a7600 


51 


14a0633 


21 


6.8066 


51 


40.1388 


22 


32.1851 


52 


148.2962 


22 


7.4691 


52 


41.7288 


58 


34.7131 


53 


153.6018 


23 


8.1636 


53 


43.3487 


24 


37.3333 


64. 


169.0000 


24 


8.8889 


54 


45.0000 


25 


40.0460 


55 


164.4906 


26 


9.6455 


65 


46.6820 


26 


42.8618 


56 


170.0740 


26 


ia4321 


56 


48.3950 


27 


46.7500 


57 


176.7500 


27 


11.2500 


67 


60.1388 


28 


48.7407 


58 


181.5186 


28 


12.0587 


68 


61.9185 


29 


61.8240 


59 


187.3796 


29 


12.9782 


69 


58.7191 


•30 


55.0000 


60 


193.3333 


30 


13.8889 


60 

H 


55.6666 



BASE 24— SLOPE 3 to 1. 





Add. 


1 


Add. 


if. of 
Hts. 


Deduct. 


if. of 
Hts. 


Deduct. 


s 




X 




O 




Q 


• 


1 


.6000 


31 


67.1666 


1 


.0185 


31 


17.7963 


2 


1.1111 


32 


71.1111 


2 


.0740 


32 


18.9630 


3 


1.8333 


33 


76.1666 


3 


.1667 


33 


20.1666 


4 


2.6666 


34 


79.3333 


4 


.2961 


34 


21.4074 


5 


3.6111 


35 


83.6111 


6 


.4628 


36 


22.6851 


6 


4.6666 


36 


88.0000 


6 


.6667 


36 


24.0000 


7 


5.8333 


37 


92.5000 


7 


.9074 


37 


26.3618 


8 


7.1111 


38 


97.1111 


8 


1.1862 


38 


26.7407 


9 


8.6000 


39 


101.8333 


9 


1.5000 


39 


28.1666 


10 


10.0000 


40 


106.6666 


10 


1.8618 


40 


29.6296 


11 


11.6111 


41 


111.6111 


11 


2.2407 


41 


31.1296 


12 


13.3333 


42 


116.6666 


12 


2.6667 


42 


32.6666 


13 


15.1666 


43 


121.8333 


13 


3.1296 


43 


34.2405 


14 


17.1111 


44 


127.1111 


14 


3.6296 


44 


35.8518 


15 


19.1666 


45 


132.5000 


16 


4.1667 


45 


37.5000 


16 


21.3333 


46 


138.0000 


16 


4.7407 


46 


39.1851 


17 


23.6111 


47 


143.6111 


17 


6.3618 


47 


40.9074 


18 


26.0000 


48 


149.3333 


18 


6.0000 


48 


42.6666 


19 


28.6000 


49 


155.1666 


19 


6.6862 


49 


44.4629 


20 


31.1111 


60 


162.1111 


20 


7.4074 


50 


46.2963 


21 


33.8333 


51 


167.1666 


21 


8.1667 


51 


49.1666 


22 


36.6666 


52 


173.3333 


22 


8.9^9 


62 


50.0740 


23 


39.6111 


53 


179.6111 


23 


9.7962 


53 


52.0186 


24 


42.6666 


54 


186.0000 


24 


10.6667 


54 


64.0000 


25 


45.8333 


55 


192.6000 


26 


11.5741 


65 


66.0184 


26 


49.1111 


56 


199.1111 


26 


12.5184 


56 


68.0640 


27 


52.5000 


57 


206.8333 


27 


13.6000 


57 


60.1666 


28 


56.0000 


58 


212.6666 


28 


14.6185 


58 


62.2962 


29 


^.6111 


59 


219.6111 


29 


16.5739 


59 


64.4629 


30 


63.3333 


60 


226.6666 


30 


16.6667 


60 


66.6666 



J 





( li- ) 




BASE 25-SLOPE i to 1. 


-a 


* 


I 




".a 




"Sa 




•8 


Add. 


•s 


Add. 


SaSJ 


Deduct. 


■sn 


Deduct. 


X 




X 




a 




o 




1 


.4676 


31 


18.8009 


1 


.0015 


31 


1.4830 


2 


.9444 


32 


19.6555 


2 


.0062 


32 


1.6802 


3 


1.4305 


33 


20.3194 


3 


.0139 


33 


1.6805 


4 


1.9259 


34 


21.0925 


4 


.0246 


34 


1.7839 


5 


2.4305 


35 


21.8760 


6 


.0385 


35 


1.8904 


6 


2.9444 


36 


22.6666 


6 


.0555 


36 


2.0000 


7 


3.4676 


37 


23.4676 


7 


.0766 


37 


2.1126 


8 


4.0000 


38 


24.2777 


8 


.0988 


38 


2.2284 


9 


4.5416 


39 


25.0972 


9 


.1250 


39 


2.3472 


10 


5.0926 


40 


25.9259 


10 


.1543 


40 


2.4691 


11 


5.6528 


41 


26.7639 


11 


.1867 


41 


2.5941 


12 


6.2222 


42 


27.6111 


12 


.2222 


42 


2.7222 


13 


6.8009 


43 


28.4676 


13 


.2608 


43 


2.8545 


14 


7.3888 


44 


29.3333 


14 


.3025 


44 


2.9876 


15 


7.9861 


45 


30.2083 


15 


.3472 


45 


3.1250 


16 


8.5925 


46 


31.0925 


16 


.3951 


46 


3.2654 


17 


9.2083 


47 


31.9861 


17 


.4460 


47 


3.4089 


18 


9.8333 


48 


32.8888 


18 


.6000 


48 


3.5666 


19 


10.4675 


49 


33.8009 


19 


.5671 


49 


3.7052 


20 


11.1111 


50 


34.7222 


20 


.6173 


50 


3.8680 


21 


11.7639 


51 


35.6528 


21 


.6806 


51 


4.0139 


22 


12.4259 


52 


36.5926 


22 


.7469 


52 


4.1728 


23 


13.0972 


53 


37.5416 


23 


.8163 


53 


4.3349 


24 


13.7777 


54 


38.5000 


24 


.8889 


54 


45000 


25 


14.4676 


55 


39.4676 


25 


.9647 


56. 


4.6682 


26 


15.1666 


56 


40.4444 


26 


1.0432 


56 


4.8395 


27 


15.8760 


57 


41.4305 


27 


1.1250 


67 


6.0139 


28 


16.5925 


58 


42.4259 


28 


1.2099 


58 


5.1913 


29 


17.3194 


59 


4.3.4305 


29 


1.2978 


59 


6.3719 


30 18.0555 

• 


60 


44.4444 


30 


1.3889 


60 6.6555 



( lii- ) 
BASE 26— SLOPE i to 1. 



1 


Add. 


•r 


w 

Add. 


1 


Deduct. 


2« 

31 


Deduct. 


1 


.4722 


31 


23.2600 


.0031 


2.9660 


2 


.9629 


32 


242962 


2 


.0123 


82 


3.1605 


3 


1.4722 


33 


25.3611 


3 


.0278 


33 


3.6111 


4 


2.0000 


34 


26.U4A 


4 


.0494 


34 


3.6679 


5 


2.6462 


36 


27.5463 


5 


.0772 


36 


3.7808 


6 


3.1111 


36 


28.6666 


6 


.1111 


36 


4.0000 


7 


3.6944 


37 


29.8056 


7 


.1612 


37 


4.2253 


8 


4.2963 


38 


30.9629 


8 


.1976 


«JO 


4.4568 


9 


4.916& 


39 


32.1388 


9 


.2500 


39 


4.6944 


10 


5.6665 


40 


33.3333 


10 


.3086 


40 


4.9383 


11 


6.2130 


41 


34.6463 


11 


.3734 


41 


5.1883 


12 


6.8888 


42 


36.7777 


12 


.4444 


42 


5.4444 


13 


7.5833 


43 


37.0277 


13 


.6216 


43 


5.7076 


14 


8.2962 


44 


38.2962 


14 


.6049 


44 


6.9763 


15 


9.0277 


46 


39.6833 


16 


.6944 


46 


6.2500 


16 


9.7777 


46 


40.8888 


16 


.7901 


46 


6.5306 


17 


10.5462 


47 


42.2129 


17 


.8920 


47 


6.8179 


18 


11.3333 


48 


43.6556 


18 


1.0000 


48 


7.1111 


19 


12.1388 


49 


44.9166 


19 


1.1142 


43 


7.4104 


20 


12.9629 


60 


4&2963 


20 


1.2346 


60 


7.7160' 


21 


13.8055 


51 


47.6944 


21 


1.3611 


51 


8.0277 


22 


14.6666 


62 


^.1111 


22 


1.4938 


m 


8.3456 


23 


15.5462 


53 


50.6462 


23 


1.6327 


53 


8.669T 


24 


16.4444 


54 


52.0000 


24 


1.7778 


64 


9.0000 


25 


17.3611 


56 


53.4722 


26 


1.9296 


55 


9.3364 


26 


18.2962 


56 


54.9629 


26 


2.0864 


56 


9.6790 


27 


19.2500 


57 


56.4722 


27 


2.2600 


57 


10.0277 


28 


20.2222 


58 


58.0000 


28 


2.4197 


68 


10.3827 


29 


21.2129 


69 


69.6462 


29 


2.5956 


59 


10.7438 


30 


22.2222 


60 


61.1111 


30 


2.7778 


60 


11.1111 





(liii 


•) 






BASE 26 SLOPE f to 1. 




X 

1 


Add. 


t 


Add. 


Dif.of 
Hts. 


Deduct. 


Dif. of 
Hts. 


DedDct. 


.4768 


31 


27.6991 


1 


.0046 


31 


4.4490 


2 


.9815 


32 


29.0370 


2 


.0185 


32 


4.7407 


3 


1.6139 


33 


30.4028 


3 


.0416 


33 


6.0416 


4 


2.0741 


34 


31.7962 


4 


.0740 


34 


5.3618 


6 


2.6619 


36 


33.2176 


6 


.1157 


36 


6.6712 


6 


3.2777 


36 


346666 


6 


.1667 


36 


6.0000 


7 


3.9212 


37 


36.1434 


7 


.2268 


37 


6.3379 


8 


4.5926 


38 


37.6481 


8 


.2963 


38 


6.6851 


9 


5.2916 


39 


39.1806 


9 


.3760 


39 


7.0416 


10 


6.0184 


40 


40.7407 


10 


.4630 


40 


7.4074 


11 


6.7732 


41 


42.3288 


11 


.6602 


41 


7.7824 


12 


7.6555 


42 


43.9444 


12 


.6667 


42 


8.1666 


13 


8.3667 


43 


45.5879 


13 


.7824 


43 


8.6612 


14 


9.2036 


44 


47.2592 


14 


.9074 


44 


8.9629 


15 


10.0603 


45 


48.9583 


15 


1.0417 


46 


9.3760 


16 


10.9620 


46 


50.6851 


16 


1.1862 


46 


9.7962 


17 


11.8842 


47 


52.4397 


17 


1.3379 


47 


10.2268 


18 


12.8333 


48 


64.2222 


18 


1.6000 


48 


10.6666 


19 


13.8101 


49 


66.0323 


19 


1.6713 


49 


11.1167 


20 


14.8147 


60 


67.8704 


20 


1.8518 


60 


11.6740 


21 


15.8472 


51 


^.7361 


21 


2.0417 


61 


12.0416 


22 


16.9074 


52 


61,6296 


22 


2.2407 


52 


12.6185 


23 


17.9953 


63 


63.6608 


23 


2.4491 


63 


13.0046 


24 


19.1111 


54 


66.5000 


24 


2.6667 


54 


13.6000 


26 20.2646 


66 


67.4768 


26 


2.8935 


65 


14.0046 


26 21.4258 


66 


69.4814 


26 


3.1296 


56 


14.5185 


27 122.6250 


67 


71.6139 


27 


3.3760 


57 


15.0416 


28 23.8518 


68 


73.6741 


28 


3.6296 


58 


15.5740 


29 :25.1064 


69 


76.6619 


29 


3.8935 


69 


16.1167 


30 126.3888 


60 


77.7777 


30 


4.1667 


60 


16.6666 





(liv 


.) 




BASE 26— SLOPE 1 to 1. 




} 


Add. 


•a, 

•s 


Add. 


(M 

Z^ 

SW 


Deduct. 




Deduct. 


33 




a 




a 
1 




Q 




1 


.4814 


31 


32.1481 


.0062 


31 


6.9321 


2 


1.0000 


32 


33.7777 


2 


.0247 


32 


6.3210 


3 


1.5655 


33 


35.4444 


3 


.0655 


33 


6.7222 


4 


2.1481 


34 


37.1481 


4 


.0988 


34 


7.1358 


5 


2.7777 


36 


38.8888 


5 


.1543 


36 


7.5617 


6 


3.4444 


36 


40.6666 


6 


.2222 


36 


8.0000 


7 


4.1481 


37 


42.4814 


7 


.3026 


37 


8.4506 


8 


4.8888 


38 


44.3333 


8 


.3951 


38 


8.9136 


9 


6.6666 


39 


46.2222 


9 


.5000 


39 


9.3888 


10 


6.4814 


40 


48.1481 


10 


.6173 


40 


9.8766 


11 


7.3333 


41 


60.1111 


11 


.74^ 


41 


10.3766 


12 


8.2222 


42 


62.1111 


12 


.8889 


42 


10.8888 


13 


9.1481 


43 


64.1481 


13 


1.0432 


43 


11.4136 


14 


10.1111 


44 


56.2222 


14 


1.2099 


44 


11.9506 


15 


11.1111 


45 


68.3333 


16 


1.3889 


46 


12.5000 


16 


12.1481 


46 


60.4814 


16 


1.5802 


46 


13.0617 


17 


13.2222 


47 


62.6666 


17 


1.7839 


47 


13.6368 


18 


14.3333 


48 


64.8888 


18 


2.0000 


48 


14.2222 


19 


16.4814 


49 


67.1481 


19 


2.2284 


49 


14.8209 


20 


16.6666 


60 


69.4444 


20 


2.4691 


60 


16.4321 


21 


17.8888 


61 


71.7777 


21 


2.7222 


61 


16.0655 


22 


19.1481 


52 


74.1481 


22 


2.9876 


62 


16.6913 


23 


20.4444 


53 


76.6666 


23 


3.2654 


63 


17.3395 


24 


21.7777 


54 


79.0000 


24 


3.5556 


64 


18.0000 


25 


23.1481 


55 


81.4814 


26 


3.8530 


56 


18.6728 


26 


24,5555 


66 


84.0000 


26 


4.1728 


66 


19.3580 


27 


26.0000 


57 


86.5655 


27 


4.5000 


67 


20.0556 


28 


27.4814 


68 


89.1481 


28 


4.8395 


58 


20.7654 


29 


29.0000 


59 


91.7777 


29 


6.1913 


69 


21.4876 


30 


30.5555 


60 


94.4444 


30 


6.5555 


60 


22.2222 







(Iv. 


) 








BASt; 25— SLOPE 1^ to 1. 


■ 


i 


Add. 


1 


Add. 


S3 33 


Deduct. 




Dednct. 


1 




B 




O 




31 




.4860 


3i 


36.6973 


1 


.0077 


7.4151 


2 


1.0185 


32 


38.6185 


2 


.0309 


32 


7.9012 


3 


1.6972 


33 


40.4861 


3 


.0694 


33 


8.4027 


4 


2.2222 


34 


42.5000 


4 


.1234 


34 


8.9197 


5 


2.8934 


35 


44.6601 


5 


.1928 


35 


9.4521 


6 


3.6111 


36 


46.6666 


6 


.2778 


36 


10.0000 


7 


4.3750 


37 


48.8192 


7 


.3781 


37 


10.5632 


8 


5.1861 


38 


51.0185 


8 


.4938 


38 


11.1419 


9 


6.0416 


39 


53.2368 


9 


.6250 


39 


11.7361 


10 


6.9444 


40 


55.5566 


10 


. .7716 


40 


12.34'56 


11 


7.8936 


41 


57.8935 


11 


.9336 


41 


12.9706 


12 


8.8888 


42 


60.2777 


12 


1.1111 


42 


13.6111 


13 


9.9305 


43 


62.7083 


13 


1.3040 


43 


14.2680 


14 


11.0185 


44 


65.1852 


14 


1.5123 


44 


14.9382 


15 


12.1527 


45 


67.7083 


15 


1.7361 


45 


15.6250 


16 


13.3333 


46 


70.2777 


16 


1.9763 


46 


16.3271 


17 


14.5601 


47 


72.8935 


17 


2.2299 


47 


17.0447 


18 


15.8333 


48 


75.6665 


18 


2.5000 


48 


17.7777 


19 


17.1527 


49 


78.2638 


19 


2.7856 


49 


18.5262 


20 


18.5184 


50 


81.0185 


20 


3.0864 


60 


19.2901 


21 


19.9305 


61 


83.8192 


21 


3.4028 


51 


20.0694 


22 


21.3888 


52 


86.6666 


22 


3.7346 


52 


20.8641 


23 


22.8935 


53 


89.6601 


23 


4.0818 


53 


21.6743 


24 


24.4444 


54 


92.6000 


24 


4.4444 


54 


22.5000 


25 


26.0416 


55 


95.4861 


25 


4.8228 


55 


23.3410 


26 


27.6851 


56 


98.5185 


26 


6.2160 


56 


24.1975 


27 


29.3750 


57 


101.5973 


27 


5.6250 


67 


25.0694 


28 


31.1111 


68 


104.7222 


28 


6.0494 


68 


25.9567 


29 


32.8935 


59 


107.8935 


29 


6.4891 


59 


26.8695 


30 


34.7222 


60 


111.1111 


30 


6.9444 


60 


27.7777 







( Ivi. ) 






BASE 25 SLOPE 


1^ to 1. 


1 


Add. 


•s 


Add. 


533 


Dednct. 


".a 

SB 


Deduct. 


s 




a: 




a 




Q 




1 


.4906 


31 


4i:0464 


1 


.0092 


31 


8.8981 


2 


1.0370 


32 


43.2592 


2 


.0370 


32 


9.4816 


3 


1.6389 


33 


45.5278 


3 


.0833 


33 


10.0833 


4 


2.2964 


34 


47.8518 


4 


.1480 


34 


10.7037 


6 


3.0092 


35 


50.2314 


5 


.2313 


36 


11.3425 


6 


3.7777 


36 


62.6666 


6 


.3333 


36 


12.0000 


7 


4.6016 


37 


55.1572 


7 


.4537 


37 


12.6759 


8 


5.4814 


38 


57.7037 


8 


.5926 


38 


ia3703 


9 


6.4166 


39 


60.3055 


9 


.7500 


39 


14.0833 


10 


7.4073 


40 


62.9629 


10 


.9259 


40 


14.8148 


11 


8.4538 


41 


65.6759 


11 


1.1203 


41 


15.5648 


12 


9.5555 


42 


69.4444 


12 


1.3333 


42 


16.3333 


13 


10.7129 


43 


71.2685 


13 


1.5648 


43 


17.1202 


14 


11.9259 


44 


74.1481 


14 


1.8148 


44 


17.9259 


15 


13.1944 


45 


77.0833 


15 


2.0833 


45 


18.7500 


16 


14.5185 


46 


80,0740 


16 


2.3704 


46 


19.5925 


17 


16.8981 


47 


83.1202 


17 


2.6759 


47 


20.4637 


18 


17.3333 


48 


86.2222 


18 


3.0000 


48 


21.3333 


19 


18.8240 


49 


89.3794 


19 


3.3^26 


49 


22.2314 


20 


20.3703 


50 


92.5926 


20 


3.7037 


50 


23.1481 


21 


21.9722 


51 


95.8611 


21 


4.0833 


61 


24.0833 


22 


23.6296 


52 


99.1851 


22 


4.4815 


62 


25.0370 


23 


25.3426 


53 


102.5647 


23 


4.8981 


63 


26.0092 


24 


27.1111 


54 


106.0000 


24 


6.3333 


54 


27.0000 


25 


28.9350 


55 


109.4906 


25 


5.7869 


55 


28.0092 


26 


30.8147 


56 


113.0370 


26 


6.2692 


66 


29.0370 


27 


32.7500 


67 


116.6389 


27 


6.7500 


57 


30.0833 


28 


34.7407 


58 


120.2962 


28 


7.2692 


68 


31.1481 


29 


36.7870 


59 


124.0092 


29 


7.7869 


59 


32.2314 


30 


38.8888 


60 


127.7777 


30 


8.3333 


60 


33.3333 





( IvH- ) 


BASE 26— SLOPE If to 1. 


•«-> 


Add. 


1 

•8 


Add. 


if. of 
Hts. 


Deduct. 


if. of 
Elts. 


Deduct. 


a 




s 




O 




Q 




1 


.4952 


31 


45.^52 


1 


.0108 


31 


10.3811 


2 


1.0556 


32 


48.0000 


2: 


.0432 


% 


11.0617 


S 


1.6806 


33 


60.5696 


3 


.0972 


33 


11.7638 


4 


2.3704 


34 


63.2037 


4 


,1728 


34 


12.4876 


6 


3.1250 


35 


56.9027 


6 


.2700 


35 


13.2380 


6 


3.9444 


36 


68.6666 


6 


.3889 


36 


140000 


7 


4.8284 


37 


61.4952 


7 


.5293 


37 


14.7885 


8 


6.7777 


38 


64.3888 


8 


.6913 


38 


16.5987 


9 


6.7916 


39 


67.3471 


9 


.8750 


39 


16.4306 


10 


7.8702 


40 


70.3703 


10 


1.0802 


40 


17.2839 


11 


9.0140 


41 


73.4583 


11 


1.3071 


41 


18.1689 


12 


10.^222 


42 


77.611] 


12 


1.6565 


^ 


19.0665 


13 


11.4962 


43 


79.8286 


13 


1.8266 


43 


19.9747 


14 


12.8333 


44 


83.1111 


14 


2.1173 


44 


20.9136 


16 


142361 


45 


86.4683 


16 


2.4305 


45 


21.8750 


16 


16.7037 


46 


89.8703 


16 


2.7664 


46 


22.8680 


17 


17.2361 


47 


93.3471 


17 


3.1219 


47 


23.8626 


18 


ia8333 


48 


96.8888 


18 


3.5000 


48 


24.8888 


19 


20.4952 


49 


100.4962 


19 


a8997 


49 


25.9367 


20 


22.2222 


60 


104.1666 


20 


4.3210 


50 


27.0061 


21 


24.0140 


61 


107.9027 


21 


47639 


61 


28.0972 


22 


25.8702 


62 


111.7037 


22 


6.2284 


62 


29.2098 


23 


27.7916 


63 


115.5696 


23 


6.7146 


63 


30.3441 


24 


29.7777 


64 


119.6000 


24 


6.2222 


64 


31.6000 


23 


31.8284 


65 


123.4962 


26 


&7616 


65 


32.6774 


26 


33.9444 


66 


127.6566 


26 


7.3024 


66 


33.8765 


27 


36.1250 


57 


131.ffi06 


27 


7.8750 


67 


35.0972 


28 


38.3704 


58 


136.8703 


28 


S.4691 


68 


36.3394 


29 


40.6806 


59 


140.1250 


29 


9.0848 


69 


37.6033 


30 


43.0556 


60 


144.4444 


30 


9.7222 


60 
I 


38.8888 



( Iviii. ) 




BASE 26— SLOPE 2 to 1. 


•8 


Add. 


1 

•8 


Add. 


if. of 
Hts. 


Deduct 


if. of 
Hts. 


Dedmct. 


a 




n 




O 




« 




1 


.6000 


31 


49.9444 


1 


.0123 


31 


11.86^ 


2 


1.0740 


32 


82.7407 


2 


.0494 


32 


12.6419 


3 


1.7222 


33 


66.6111 


3 


.1111 


33 


13.4444 


4 


2.4444 


34 


68.6656 


4 


.1976 


34 


142716 


6 


3.2407 


36. 


61.6740 


6 


.3086 


36 


15.1234 


6 


4.1111 


96 


64.6666 


6 


.,4444 


36 


laoooo 


7 


6.0666 


37 


67.8333 


7 


.6049 


37 


16.9012 


8 


6.0740 


38 


71.0740 


8 


.7901 


38 


17.8271 


9 


7.1666 


39 


74.3888 


9 


1.0000 


39 


18.7777 


10 


8.3333 


40 


77.7777 


10 


1.2346 


40 


19.7630 


11 


9.6741 


41 


81.2407 


11 


1.4938 


41 


20.7530 


12 


10.8888 


^ 


86.7777 


12 


1.7778 


42 


21.7777 


13 


12.2777 


43 


88.3888 


13 


2.0864 


43 


22.8271 


14 


13.7407 


44 


92.0740 


14 


2.4197 


44 


23.9012 


16 


14.2777 


46 


95.8333 


16 


2.7778 


46 


26.0000 


16 


16.8888 


46 


99.6666 


16 


3.1606 


46 


26.1234 


17 


18.6741 


47 


103.5740 


17 


3.5679 


47 


27.2716 


18 


20.3333 


48 


107.6565 


18 


40000 


48 


28.4444 


19 


22.1666 


49 


111.6111 


19 


4.4568 


^ 


29.6420 


20 


240740 


60 


115.7407 


20 


49382 


50 


30.8642 


21 


26.0556 


61 


119.9444 


21 


5.4444 


61 


32.1111 


22 


28.1111 


62 


1242222 


22 


6.9763 


52 


33.3827 


23 


30.2407 


63 


128.6740 


23 


&6309 


63 


34.6790 


24 


32.4444 


64 


133.0000 


24 


7.1111 


64 


36.0000 


25 


34.7222 


66 


137.6000 


26 


7.7160 


66 


37.3466 


26 


37.0740 


66 


142.0740 


26 


8.3467 


66 


38.7160 


27 


39.5000 


67 


146.7222 


27 


9.0000 


67 


40.1111 


28 


42.0000 


68 


161.4444 


28 


9.6790 


58 


41.6308 


29 


44.5740 


69 


156.2407 


29 


10.3827 


69 42.9763 


30 


47.2222 


60 


161.1111 1 30 


11.1111 


60 44.4444 







(liz.) 


BASE 25— SLOPE 1^ to 1. 


1. 
•s 


Add. 


1 


Add. 


if. of 
Hts. 


Dednct. 


if. of 


Deduct. 


K 




K 




O 




Q 




1 


.5092 


31 


58.8^5 


1 


.0154 


31 


148302 


2 


1.1111 


32 


62.2222 


2 


.0617 


32 


16.8013 


3 


1.8065 


33 


66.6944 


3 


.1389 


33 


16.8056 


4 


2.5926 


34 


69.2692 


4 


.2468 


34 


17.8306 


5 


3.4722 


36 


72.9166 


5 


.3857 


36 


18.9043 


6 


4.4444 


36 


76.6666 


6 


.5556 


36 


20.0000 


7 


5.5092 


37 


80.5092 


7 


.7561 


37 


21.1265 


8 


6.6666 


38 


844444 


8 


.9876 


38 


22.2839 


9 


7.9166 


39 


88.4722 


9 


1.2500 


39 


23.4722 


10 


9.2592 


40 


92.8386 


10 


1.64^ 


40 


246913 


11 


10.6944 


41 


96.8055 


11 


1.8673 


41 


26.9413 


12 


12.2222 


^ 


101.1111 


12 


2.2222 


42 


27.2222 


13 


13.8425 


43 


105.6092 


13 


2.6080 


43 


28.5360 


14 


15.5555 


44 


110.0000 


14 


a0247 


44 


29:8765 


15 


17.3611 


46 


1145833 


15 


3.4722 


45 


31.2600 


16 


19.2592 


46 


119.2592 


16 


3.9506 


46 


32.6543 


17 


21.2500 


47 


124.0277 


17 


44599 


47 


340895 


18 


23.3333 


48 


128.8888 


18 


5.0000 


48 


36.6665 


19 


25.6092 


49 


133.8^6 


19 


5.5710 


49 


37.0524 


20 


27.7777 


50 


138.8888 


20 


&1728 


50 


38.6802 


21 


30.1388 


51 


144.0277 


21 


6.8066 


51 


40.1388 


22 


32.6926 


52 


1^.2692 


22 


7.4691 


62 


41.7283 


23 


35.1388 


53 


1545833 


23 


8.1636 


53 


43.3487 


24 


37.7777 


64 


160.0000 


24 


8.8889 


54 


46.0000 


25 


40.6092 


55 


165.6092 


25 


9.6455 


55 


46.6820 


26 


43.3333 


66 


171.1111 


26 


10.4321 


66 


48.3960 


27 


46.2500 


67 


176.8055 


27 


11.2500 


57 


60.1388 


28 


49.2692 


58 


182.5926 


28 


12.0587 


68 


51.9136 


29 


62.3611 


59 


188.4722 


29 


12.9782 


59 


53.7191 


30 


65.6556 


60 


1944444 


30 


13.8889 1 60 


65.6555 



• 


(1x0 




BASE 25— SLOPE 3 to 1. 


1 


Add. 


} 


Add. 




Deduct. 




Bednct. 


1 




m 




1 




Q 




.5185 


31 


67.7407 


.0185 


31 


17.7963 


2 


1.1482 


32 


71.7037 


2 


.0740 


32 


18.9630 


3 


1.8888 


33 


75.7777 


8 


.1667 


33 


2ai666 


4 


2.7407 


34 


79.9629 


4 


.2968 


34 


21.4074 


5 


3.7037 


35 


84.2692 


5 


.4628 


36 


22.6851 


6 


4.7777 


36 


8a6666 


6 


.6667 


36 


24.0000 


7 


5.9629 


37 


93.1850 


7 


.9074 


37 


25.3518 


8 


7.2592 


38 


97.8148 


8 


1.1852 


38 


2a7407 


9 


8.6666 


39 


102.5655 


9 


1.5000 


39 


28.1666 


10 


10.1851 


40 


107.4074 


10 


1.8518 


40 


29.6296 


11 


11.8148 


41 


112.3703 


11 


2.2407 


41 


31.1296 


12 


13.5555 


^ 


117.4444 


12 


2.6667 


4i 


32.6666 


13 


16.4074 


43 


122.6296 


13 


3.1296 


43 


34.2407 


14 


17.3703 


44 


127.9258 


14 


3.6296 


44 


35.8518 


15 


19.4444 


45 


133.3333 


15 


41667 


45 


37.6000 


16 


21.^96 


46 


138.8518 


16 


47407 


46 


39.1861 


17 


23.9259 


47 


1444814 


17 


5.3518 


47 


40.9074 


18 


26.3333 


48 


150.2222 


18 


6.0000 


48 


^6666 


19 


28.8518 


49 


156.0740 


19^ 


6.6852 


49 


44.4629 


20 


31.4815 


50 


1&LOS70 


20 


7.4074 


50 


46.2963 


21 


34.2222 


51 


168.1111 


21 


8.1667 


51 


49.1666 


22 


37.0740 


52 


1742962 


22 


8.9629 


52 


50.0740 


23 


40.0370 


53 


180.5926 


23 


9.79^ 


53 


62.0185 


24 


43.1111 


54 


187.0000 


24 


10.6667 


54 


64.0000 


25 


46.2962 


55 


193.5185 


25 


11.5741 


55 


66.0184 


26 


49.5926 


56 


200.1481 


26 


12.5184 


66 


68.0740 


27 


53.0000 


57 


206.8888 


27 


13.6000 


57 


60.1666 


28 


56.5185 


58 


213.7407 


28 


14.5185 


68 


62.2962 


29 


60.1482 


59 


220.7037 


29 


15.5741 


69 


64.4629 


30 


63.8888 


60 


227.7777 


30 


16.6667 


60 


66.6666 





(Ixi.) 


BASE 26— SLOPE J to 1. 


} 


Add. 


1 


Add. 




Deduct. 


"S 


Deduct. 


» 




Si 




1 




o" 




1 


.4861 


31 


19.3750 


.0015 


31 


1.4830 


2 


.9814 


32 


20.1481 


2 


.0062 


32 


1.6802 


3 


1.4861 


33 


20.9305 


3 


.0139 


33 


1.6806 


4 


2,0000 


34 


21.7222 


4 


.0246 


34 


1.7839 


5 


2.5231 


35 


22.5231 


5 


.0385 


35 


1.8904 


6 


3.0556 


36 


23.3333 


6 


.0555 


36 


2.0000 


7 


3.6972 


37 


24.1527 


7 


.0766 


37 


2.1126 


8 


4.1481 


38 


24.9814 


8 


.0988 


38 


2.2284 


9 


47083 


39 


25.8194 


9 


.1250 


39 


2.3472 


10 


5.2777 


40 


26.6666 


10 


.1543 


40 


2.4691 


11 


6.8664 


41 


27.5231 


11 


.1867 


41 


2.5941 


12 


6.4444 


42 


28.3888 


12 


.2222 


42 


2.7222 


13 


7.0416 


43 


29.2638 


13 


.2608 


43 


2.8545 


14 


7.6481 


44 


30.1481 


14 


.3025 


44 


2.9876 


15 


8.2638 


45 


31.0416 


15 


.3472 


45 


3.1250 


16 


8.8888 


46 


31.9444 


16 


.3961 


46 


3.2654 


17 


9.6231 


47 


32.8564 


17 


.4460 


47 


3.4089 


18 


10.1666 


48 


33.7777 


18 


.6000 


48 


3.6666 


19 


10.8194 


^ 


34.7083 


19 


.6.571 


49 


3.7062 


20 


11.4814 


60 


36.6481 


20 


.6173 


60 


3.8580 


21 


12.1627 


51 


36.6972 


21 


.6806 


51 


4.0139 


22 


12.8333 


52 


37.5655 


22 


.7469 


52 


4.1728 


23 


13.5231 


63 


S8.£@3l 


23 


.8163 


63 


4.3349 


24 


14.2222 


64 


39.6000 


24 


.8889 


64 


46000 


25 


14.9306 


55 


40.4861 


25 


.9647 


66 


4.6682 


26 


16.6481 


56 


41.4814 


26 


1.0432 


66 


48395 


27 


16.3760 


57 


42.4861 


27 


1.1260 


67 


6.0139 


28 


17.1111 


68 


43.6000 


28 


1.2099 


68 


5.1913 


29 


17.8664 1 69 


44.6231 


29 


1.2978 


69 


6.3719 


30 


18.6111 1 60 


45.5556 


30 


1.3889 


60 


5.5655 





( I'^ii- ) 




BASE 26— SLOPE i to 1. 


1 


Add. 


** 


Add. 




Deduct. 




Dednct. 


n 
1 




n 

31 




1 




Q* 




.4907 


23.8241 


.0031 


31 


2.9660 


2 


1.0000 


32 


24.8888 


2 


.0123 


32 


3.1606 


8 


1.6277 


33 


25.9722 


3 


.0278 


33 


3.3611 


4 


2.0741 


34 


27.0740 


4 


.0494 


34 


8.5679 


6 


2.6388 


36 


28.1944 


5 


.0772 


36 


3.7808 


6 


a2222 


36 


29.3333 


6 


.1111 


36 


4.0000 


7 


3.8240 


37 


30.4908 


7 


.1512 


37 


4.2253 


8 


4.4444 


38 


31.6666 


8 


.1976 


38 


4.4668 


9 


6.0833 


39 


32.8611 


9 


.2600 


39 


4.6944 


10 


6.7407 


40 


34.0740 


10 


.3086 


40 


4.9383 


11 


6.4166 


41 


35.3056 


11 


.3734 


41 


5.1883 


12 


7.1111 


42 


36.5665 


12 


.4444 


^ 


6.4444 


13 


7.8240 


43 


37.8240 


IS 


.6216 


43 


6.7067 


14 


8.5665 


44 


39.1111 


14 


.6049 


44 


5.9763 


16 


9.3054 


45 


40.4166 


16 


.6944 


46 


6.2600 


16 


10.0740 


46 


41.7407 


16 


.7901 


46 


6.6308 


17 


10.8611 


47 


43.0833 


17 


.8920 


47 


&8179 


18 


11.6666 


48 


44.4444 


18 


1.0000 


48 


7.1111 


19 


12.4907 


49 


45.8240 


19 


1.1142 


49 


7.4105 


20 


13.3333 


50 


47.2222 


20 


1.2346 


50 


7.7160 


21 


14.1944 


51 


48.6388 


21 


1.3611 


61 


8.0277 


22 


15.0740 


62 


60.0740 


22 


1.4938 


52 


8.3456 


23 


15.9722 


63 


61.5277 


23 


1.6327 


63 


8.6697 


24 


16.8888 


64 


53.0000 


24 


1.7778 


54 


9.0000 


25 


17.8240 


66 


64.4907 


2g 


1.9296 


66 


9.3364 


26 


18.7777 


56 


56.0000 


26 


2.0864 


66 


9.6790 


27 


19.7500 


57 


57.6277 


27 


2.2600 


57 


10.0277 


28 


20.7407 


58 


69.0740 


28 


2.4197 


58- 


10.3827 


29 


21.7500 


59 


60.6388 


29 


2.6956 


59 


10.7438 


30 


22.7777 


60 


62.2222 


30 


2.7778 


60 


11.1111 







( Ixiii. ) 




BASE 26— SLOPE f to 1. 


1 


* 

Add. 


1 


Add. 




Deduct. 




Deduct. 


» 




» 




1 




a" 




1 


.^53 


31 


28.2732 


.0046 


31 


44490 


2 


1.0185 


32 


29.6296 


2 


.0186 


32 


47407 


3 


1.5694 


33 


31.0139 


3 


.0416 


33 


6.0416 


4 


2.1481 


34 


32.4258 


4 


.0740 


34 


5.3518 


5 


2.7545 


35 


33.8657 


5 


.1167 


35 


5.6712 


6 


3.3888 


36 


35.3333 


6 


.1667 


36 


6.0000 


7 


4.0508 


37 


36.8287 


7 


,2268 


37 


6.3379 


8 


4.7407 


38 


38.3518 


8 


.2963 


38 


6.6851 


9 


5.4583 


39 


39.9027 


9 


.3750 


39 


7.0416 


10 


6.2036 


40 


41.4814 


10 


.4630 


40 


7.4074 


11 


&9768 


41 


43.0680 


11 


.6602 


41 


7.7824 


12 


7.7777 


42 


44.7222 


12 


.6667 


^ 


8.1666 


13 


8.6064 


43 


46.3842 


13 


.7824 


43 


8.6602 


14 


9.4629 


44 


48.0740 


14 


.9074 


44 


8.9629 


16 


10.3470 


45 


49.7916 


15 


1.0417 


45 


9.3760 


16 


11.2592 


46 


51.6370 


16 


1.1862 


46 


9.7962 


17 


12.1990 


47 


53.3101 


17 


1.3379 


47 


10.2268 


18 


13.1666 


48 


56.1111 


18 


1.6000 


48 


10.6666 


19 


141620 


49 


56.9397 


19 


1.6713 


49 


11.1157 


20 


15.1851 


50 


68.7962 


20 


1.8518 


60 


11.6740 


21 


16.2360 


51 


60.6806 


21 


2.0417 


51 


12.0416 


22 


17.3147 


52 


62.6926 


22 


2.2407 


62 


12.6185 


23 


18.4213 


53 


64.5322 


23 


2.4491 


63 


13.0046 


24 


19.5555 


54 


66.5000 


24 


2.6667 


64 


13.5000 


25 


20.7175 


55 


68.4952 


25 


2.8936 


56 


14.0046 


26 


21.9073 


56 


70.6186 


26 


3.1296 


56 


14.6185 


27 


23.1250 


57 


72.6694 


27 


3.3760 


57 


15.0416 


28 


243703 


58 


74.6481 


28 


3.6296 


68 


15.5740 


29 


25.6435 


59 


76.7646 


29 


a8935 


69 


16.1167 


30 


26.9444 


60 


78.8888 


30 416671 


60 


1&6666 





( l«v. ) 


BASE 26— SLOPE 1 to 1. 


1 

•8 


Add. 


1 


Add. 


"a 


• 

Deduct. 


3^ 


Deduct. 


M 




a 




fi" 




fi 




1 


.6000 


81 


32.7222 


1 


.0062 


31 


6.9321 


2 


1.0370 


82 


84.8703 


2 


.0247 


82 


6.821U 


8 


1.6111 


38 


86.0555 


8 


.0566 


33 


6.7222 


4 


2.2222 


84 


37.7777 


4 


.0988 


84 


7.1858 


6 


2.8704 


86 


89.5370 


5 


.1643 


85 


7.5617 


6 


3.6656 


36 


41.3388 


6 


.2222 


86 


8.0000 


7 


4.2777 


87 


43.1666 


7 


.8025 


37 


8.4506 


8 


6.0870 


38 


46.0370 


8 


.3951 


38 


&9135 


9 


5.8338 


89 


46.9444 


9 


.6000 


89 


9.3888 


10 


6.6666 


40 


48.8888 


10 


.6173 


40 


9.8765 


11 


7.5870 


41 


50.8704 


11 


.7469 


41 


10.8766 


12 


8.4444 


^ 


62.8888 


12 


.8889 


42 


10.8888 


13 


9.3888 


43 


64.9444 


18 


1.0432 


48 


11.4135 


14 


10.8703 


44 


67.0370 


14 


1.2099 


44 


11.9506 


16 


11.8888 


45 


59.1666 


15 


1.3889 


45 


12.6000 


16 


12.4444 


46 


61.8333 


16 


1.6802 


46 


13.0617 


17 


13.6370 


47 


63.5870 


17 


1.7839 


47 


13.6358 


18 


14.6666 


48 


65.7777 


18 


2.0000 


48 


14.2222 


19 


16.8333 


49 


68.0565 


19 


2.2284 


^ 


14.8209 


20 


17.0370 


50 


70.8703 


20 


2.4691 


60 


15.4321 


21 


18.2777 


61 


72.7222 


21 


2.7222 


61 


16.0556 


22 


19.6565 


52 


76.1111 


22 


2.9876 


52 


16.6913 


23 


20.8704 


63 


77.5870 


23 


3.2654 


63 


17.3395 


24 


22.2222 


54 


80.0000 


24 


3.6656 


54 


18.0000 


25 


23.6)11 


56 


82.6000 


25 


3.8680 


56 


18.6728 


26 


26.0370 


66 


85.0870 


26 


4.1728 


66 


19.3580 


27 


2&6000 


57 


87.6111 


27 


46000 


67 


20.0556 


28 


28.0000 


68 


90.2222 


28 


4.8395 


58' 


20.7664 


29 


29.5370 


69 


92.8704 


29 


5.1913 


59 


21.4876 


30 


31.1111 


60 


96.5556 


30 


6.6566 60 


22.2222 







( Ixv. ) 


" 


BASE 26— SLOPE IJ to L 


1 


Add. 


1 


Add. 


"a 


Deduct. 


an 


Deduct. 


n 




K 


. 


o" 




O*^ 




1 


.5046 


31 


37.1713 


1 


.0077 


31 


7.4151 


2 


1.0555 


32 


39.1111 


2 


.0309 


32 


7.9012 


3 


1.6528 


33 


41.0972 


3 


.0694 


33 


&4027 


4 


2.2963 


34 


43.1296 


4 


.1234 


34 


8.9197 


5 


2.9859 


35 


45.2083 


6 


.1928 


36 


9.4621 


6 


3.79912 


36 


47.3333 


6 


,2778 


36 


10.0000 


7 


45045 


87 


49.5046 


7 


.3781 


37 


10.5632 


8 


5.3333 


38 


61.7222 


8 


.4938 


38 


11.1419 


9 


6.2083 


39 


53.9860 


9 


.6250 


39 


11.7361 


10 


7.1296 


40 


56.2962 


10 


.7716 


40 


12.3456 


11 


8.0972 


41 


68.6628 


11 


.9336 


41 


12.9706 


12 


9.1111 


42 


61.0565 


12 


Llill 


42 


13.6111 


13 


10.1712 


43 


63.5046 


13 


1.3040 


43 


142680 


14 


11.2777 


44 


66.0000 


14 


1.5123 


44 


14.9382 


15 


12.4304 


45 


68.5416 


15 


1.7361 


45 


15.6250 


16 


13.6296 


46 


71.1296 


16 


1.9763 


46 


ia3271 


17 


14.8760 


47 


73.7639 


17 


2.2299 


47 


17.0447 


18 


16.1666 


48 


76.4444 


18 


2.5000 


48 


17.7777 


19 


17.5046 


49 


. 79.1712 


19 


2.7856 


49 


18.6262 


20 


18.8888 


50 


81.9444 


20 


3.0864 


60 


19.2901 


21 


20.3196 


61 


84.7639 


21 


a4028 


61 


20.0694 


22 


21.7962 


52 


87.6296 


22 


3.7346 


62 


20.8641 


23 


23.3195 


53 


90.6416 


23 


4.0818 


53 


21.6743 


24 


24.8888 


54 


93.5000 


24 


4.4444 


54 


22.5000 


25 


26iK)46 


65 


96.5046 


25 


4.6228 


55 


23.3410 


26 


28.1666 


66 


99.5555 


26 


5.2160 


56 


241976 


27 


29.8750 


67 


102.6528 


27 


5.6260 


57 


25.0694 


28 


31.6296 


68 


105.7962 


28 


6.0494 


58 


25.9667 


29 


33.4306 


69 


108.9860 


29 


6.4891 


59 


26.8696 


30 


86.2777 


60 


112.2222 


30 


6.9444 


60 

K 


27.7777 





( IxTi. ) 




BASE 26— SLOPE 1^ to 1. 




1 

•8 


Add. -e 


Add. 




Dednct. 




Dednct. 


M 




» 




q" 




f^ p*4 




1 


.5092 


31 


41.6204 


1 


.0092 


31 


8.8981 


2 


1.0740 


32 


43.8618 


2 


.0870 


32 


9.4815 


3 


1.6944 


33 


46.1389 


3 


.0833 


38 


10.0838 


4 


2.8704 


34 


48.4815 


4 


.1480 


84 


10.7037 


5 


3.1016 


35 


50.8796 


5 


.2313 


36 


11.3425 


6 


3.8888 


36 


53.3333 


6 


.3338 


36 


12.0000 


7 


4.7314 


37 


55.8424 


7 


.4637 


37 


12.6769 


8 


5.6296 


38 


58.4074 


8 


.5926 


38 


13.3703 


9 


6.5833 


39 


61.0277 


9 


.7500 


39 


14.0833 


10 


7.5926 


40 


63.7037 


10 


.9259 


40 


148148 


11 


8.6574 


41 


66.4362 


11 


1.1203 


41 


15.5648 


12 


9.7777 


42 


70.2222 


12 


1.3333 


42 


16.3333 


13 


10.9536 


43 


72.0648 


13 


1.6648 


48 


17.1202 


14 


12.1851 


44 


74.9629 


14 


1.8148 


44 


17.9259 


15 


13.4720 


45 


77.9166 


15 


2.0833 


45 


18.7600 


16 


148148 


46 


80.9260 


16 


2.3704 


46 


19.5925 


17 


16.2129 


47 


83.9907 


17 


2.6769 


47 


20.4537 


18 


17.6666 


48 


87.1111 


18 


3.0000 


48 


21.3833 


19 


19.1759 


49 


90.2870 


19 


3.3426 


49 


22.2314 


20 


20.7407 


50 


93.6185 


20 


3.7037 


50 


28.1481 


21 


22.8611 


51 


96.8065 


21 


40838 


61 


24.0833 


22 


24.0370 


52 


100.1481 


22 


44815 


52 


26.0370 


23 


25.7686 


53 


103.5462 


23 


48981 


53 


26.0092 


24 


27.6565 


64 


107.0000 


24 


5.3383 


54 


27.0000 


25 


29.3981 


65 


110.6092 


26 


5.7869 


66 


28.0092 


26 


31.2962 


56 


1140740 


26 


6.2592 


66 


29.0370 


27 


83.2500 


67 


117.6945 


27 


6.7600 


67 


30.0833 


28 


35.2592 58 


121.8704 


28 


7.2692 


58 


31.1481 


2d 


37.3240 69 


126.1017 


29 


7.7869 


59 


32.2314 


80 39.44441 60 


128.8888 


30 


8.3833 


60 


33.3333 





( Ixvii. ) 


BASE 26— SLOPE If to 1. 


1 


Add. 


1 
•s 


Add. 




Deduct. 


"Sri 


Dedoct. 


1 




s 




O* 




o" 




.5138 


31 


46.0694 


1 


.0108 


31 


10.3811 


2 


1.0926 


32 


48.5926 


2 


.0432 


32 


11.0617 


3 


1.7361 


33 


51.1805 


3 


.0972 


.33 


11.7638 


4 


2.4444 


34 


63.8333 


4 


.1728 


34 


12.4876 


6 


3.2176 


35 


56.6509 


5 


.2700 


35 


ia2330 


6 


4.0555 


86 


69.3333 


6 


.3889 


36 


14.0000 


7 


4.9582 


37 


62.1805 


7 


.5293 


37 


147885 


8 


5.9259 


38 


65.0926 


8 


.6913 


38 


15.5987 


9 


&9582 


39 


68.0694 


9 


.8750 


39 


16.4305 


10 


8.0555 


40 


71.1111 


10 


1.0802 


40 


17.2839 


11 


9.2176 


41 


74.2176 


11 


1.3071 


41 


18.1589 


12 


10.4444 


42 


78.3888 


12 


1.5556 


42 


19.0556 


13 


11.7361 


43 


80.^50 


13 


1.8266 


43 


19.9747 


14 


13.0926 


44 


83.9258 


14 


2.1173 


44 


20.9135 


15 


14.5138 


46 


87.2916 


15 


2.4306 


45 


21.8750 


16 


16.0000 


46 


90.7222 


16 


2.7654 


46 


22.8580 


17 


17.5508 


47 


94.2177 


17 


3.1219 


47 


23.8626 


18 


19.1666 


48 


97.7777 


18 


3.6000 


48 


24.8888 


19 


20.8472 


49 


101.4026 


19 


3.8997 


49 


26.9367 


20 


22.6926 


50 


106.0926 


20 


4.3210 


50 


27.0061 


21 


24.4026 


51 


108.8472 


21 


4.7639 


51 


28.0972 


22 


26.2777 


62 


112.6666 


22 


5.2284 


52 


29.2098 


23 


28.2177 


53 


116.5608 


23 


5.7145 


53 


30.3441 


24 


30.2222 


54 


120.51K)0 


24 


6.2222 


64 


31.5000 


25 


32.2916 


65 


124.5138 


25 


6.7516 


56 


32.6774 


26 


34.4258 


56 


128.5926 


26 


7.3024 


66 


33.8765 


27 


36.6250 


57 


182.7361 


27 


7.8750 


57 


35.0972 


28. 


.38.8888 


58 


136.9444 


28 


8.4691 


68 


36.3394 


29 


41.2176 


69 


141.2174 


29 


9.0848 


59 


37.6033 


30 


43.6111 


60 


145.6555 


30 


9.7222 


1 60 


38.8888 



( Ixviii. ) 
BASE 26— SLOPE 2 to 1. 






Add. 



1 .5185 

2 1.1111 

3 1.7777 

4 2.5185 

5 3.3333 

6 4.2222 

7 5.1851 

8 6.2222 

9 7.3333 

10 8.5185 

11 9.7777 

12 11.1111 

13 12.5185 

14 i4.oooa 

15 15.5555 

16 17.1851 

17 18.8888 

18 20.6666 

19 22.5185 
24.4444 



20 

21 
22 
23 
24 
25 

26 
27 
28 
29 
30 



26.4444 
28.5185 
30.6666 
32.8888 
35.1851 

37.5555 

40.0000 
42.5185 
45.1111 

47.7777 



•t 



Add. 



31 
32 
33 
34 
35 

36 
37 
38 
39 

40 

41 
42 
43 
44 
45 

46 
47 

48 
49 
50 

51 
52 
53 
54 
55 

56 
67 

58 
59 
60 



50.5185 
53.3333 
56.2222 
59.1851 
62.2222 

65.3333 

68.5185 
71.7777 
75.1111 
78.5185 

82.0000 
86.5555 
89.1851 
92.8888 
96.6666 

100.5185 
104.4444 
108.4444 
112.5185 
116.6666 

120.8888 
125.1851 
129.5555 
134.0000 
138.5185 

143.1111 
147.7777 
152.5185 
157.3333 
162.2222 



• a 



1 

•2 
8 

4 
6 

6 

7 

8 

9 

10 

11 
12 
13 
14 
15 

16 
17 
18 
19 
20 

21 

22. 

23 

24 

25 

26 
27 
28 
29 
SO 



Deduct. 



.0123 
.0494 
.1111 
.1975 
.3086 

A444 

• A X M. X 

.6049 

.7901 

1.0000 

1.2346 

1.4938 

1.7778 
2.0864 
2.4197 
2.7778 

3.1605 
3.5679 
4.0000 
4.4568 
4.9382 

5.4444 
5.9753 
6.5309 
7.1111 
7.7160 

8.3457 

9.0000 

9.6790 

10.3827 

11.1111 



o j5* 



32 
33 
34 
35 

36 
37 
38 
39 
40 

41 
42 
43 
44 
45 

46 
47 
48 
49 
50 

61 
52 
53 
54 
55 

56 
57 
58 
59 
60 



Deduct. 



11.8642 
12.6419 
13.4444 
14.2716 
15.1234 

16.0000 
16.9012 
17.8271 
18.7777 
19.7530 

20.7530 
21.7777 
22.8271 
23.9012 
25.0000 

26.1234 
27.2716 
28.4444 
29.6420 
30.8642 

32.1111 
33.3827 
34.6790 
36.0000 
37.3456 

38.7160 
40.1111 
41.5308 
42.9753 

AA AAAA 

XX. X X XX 







( Ixix, ) 






BASE 26— SLOPE 2J to 1. 




1 


Add. 


i 


Add. 


Dif. of 
Hts. 


Deduct. 




Deduct. 


1 


.5277 


31 


59.4166 


1 


.0154 


31 


14.8302 


2 


1.1482 


32 


62.8148 


2 


.0617 


32 


16.8016 


3 


1.8611 


33 


66.3055 


3 


.1389 


33 


16.8066 


4 


2.6666 


34 


69.8888 


4 


.2468 


34 


17.8395 


5 


3.5648 


35 


73.5648 


6 


.3857 


35 


18.9043 


6 


4.5555 


36 


77.3333 


6 


.5665 


36 


20.0000 


7 


5.6388 


37 


81.1944 


7 


.7661 


37 


21.1265 


8 


6.8148 


38 


85.1482 


8 


.9876 


38 


22.2839 


9 


8.0833 


39 


89.1944 


9 


1.2500 


39 


23.4722 


10 


9.4444 


40 


93.3333 


10 


1.5432 


40 


24.6913 


11 


10.8981 


41 


97.5648 


11 


1.8673 


41 


25.9413 


12 


12.4444 


42 


101.8888 


12 


2.2222 


42 


27.2222 


13 


14.0833 


43 


106.3055 


13 


2.6080 


43 


28.5360 


14 


15.8148 


44 


110.8148 


14 


3.0247 


44 


29.8765 


16 


17.6388 


46 


116.4166 


16 


a4722 


46 


31.2500 


16 


19.5555 


m 


120.1111 


16 


3.9506 


m 


32.6543 


17 


21.5648 


47 


124.8981 


17 


4.4699 


47 


34.0896 


18 


23.0666 


48 


129.7777 


18 


6.0000 


48 


36.6665 


19 


25.8611 


49 


134.7600 


19 


5.6710 


49 


37.0524 


20 


28.1482 


50 


139.8148 


20 


6.1728 


50 


38.6802 


21 


30.6277 


61 


144.9722 


21 


6.8066 


51 


40.1388 


22 


33.0000 


52 


150.3222 


22 


7.4691 


62 


41.7283 


23 


36.5648 


63 


165.5648 


23 


8.1636 


63 


43.3487 


24 


38.2222 


54 


161.0000 


24 


8.8889 


64 


46.0000 


25 


40.9722 


56 


166.6277 


25 


9.6466 


55 


46.6820 


26 


43.8148 


56 


172.1482 


26 


10.4321 


66 


48.3950 


27 


46.7500 


57 


177.8611 


27 


11.2500 


67 


50.1388 


28 


49.7777 


68 


183.6666 


28 


12.0687 


68 


51.9135 


29 


52.8981 


59 


189.6648 


29 


12.9782 


59 


63.7191 


30 


56.1111 


60 


196.5566 


30 


13.8889 


60 


56.6556 





( 1«. ) 


BASE 26— SLOPE 3 to 1. 


* 


Add. 


4^ 

t 


Add. 


«4M 


Deduct. 


"a 


Deduct. 


1 




as 




o 




a 




.6370 


31 


68.3148 


1 


.0186 


31 


17.7963 


2 


1.1852 


32 


72.2962 


2 


.0740 


32 


18.9630 


3 


1.9444 


33 


76.3888 


3 


.1667 


33 


20.1666 


4 


2.8148 


34 


80.6926 


4 


.2963 


34 


21.4074 


5 


3.7962 


35 


84.9074 


5 


.4628 


35 


22.6851 


6 


4.8888 


36 


89.3333 


6 


.6667 


36 


240000 


7 


6.0926 


37 


93.8703 


7 


.9074 


37 


25.3618 


8 


7.4074 


38 


98.6186 


8 


1.1862 


38 


26.7407 


9 


8.8333 


39 


103.2777 


9 


1.6000 


39 


28.1666 


10 


10.3703 


40 


108.1481 


10 


1.8518 


40 


29.^96 


11 


12.0186 


41 


113.1296 


11 


2.2407 


41 


31.1296 


12 


13.7777 


42 


118.2222 


12 


2.6667 


42 


32.6666 


13 


16.6482 


43 


123.^259 


13 


3.1296 


43 


342407 


14 


17.6296 


44 


128.7407 


14 


3.6296 


44 


36.8518 


16 


19.7222 


46 


134.1666 


16 


41667 


45 


37.6000 


16 


21.9269 


46 


139.7037 


16 


47407 


46 


39.1861 


17 


24.2407 


47 


145.3517 


17 


6.3618 


47 


40.9078 


18 


26.6666 


48 


161.1111 


18 


6.0000 


48 


42.6666 


19 


29.2037 


49 


156.9814 


19 


6.6862 


49 


44.4629 


20 


31.8618 


60 


162.9630 


20 


7.4074 


60 


46.2963 


21 


34.6111 


51 


169.0555 


21 


8.1667 


61 


49.1666 


22 


37.4816 


62 


176.2692 


22 


8.9629 


62 


60.0740 


23 


40.4630 


63 


181.5740 


23 


9.7962 


63 


62.0185 


24 


43.6566 


54 


188.0000 


24 


10.6667 


54 


54.0000 


26 


46.7592 


65 


1946370 


26 


11.6741 


56 


56.0184 


26 


60.0740 


66 


201.1861 


26 


12.6184 


66 


68.0740 


27 


53.6000 


67 


207.9444 


27 


13.5000 


57 


60.1666 


28 


57.0370 


68 


214.8148 


28 


145186 


68 


62.2962 


29 


60.6861 


69 


221.7962 


29 


15.6741 


69 


64.4629 


30 


64.4444 


60 


228.8888 

• 


30 


ia6667 


60 


66.6666 



( Ixxi. ) 




BASE 27-8LOPE 


i to 1. 


1 
•s 


Add. 


■^ 

t 


Add. 


*** 

saaa 


Deduct. 




Deduct. 


B 




a 




o 




Q 




1 


.6046 


31 


19.9490 


1 


.0016 


31 1.4830 


2 


1.0186 


32 


20.7407 


2 


.0062 


32 1.5802 


3 


1.6416 


33 


21.6416 


3 


.0139 


33 


1.6805 


4 


2.0740 


34 


22.3518 


4 


.0246 


34 


1.7839 


5 


2.6167 


36 


23.1712 


6 


.0385 


35 


1.8904 


6 


3.1666 


36 


24.0000 


6 


.0566 


36 


2.0000 


7 


3.7268 


37 


24.8379 


7 


.0766 


37 2.1126 


8 


4.2962 


38 


26.6861 


8 


.0988 


38 


2.2284 


9 


4,8760 


39 


26.6416 


9 


.1250 


39 


2.3472 


10 


6.4629 


40 


27.4074 


10 


.1643 


40 


2.4691 


11 


&0601 


41 


28.2824 


11 


.1867 


41 


2.5941 


12 


6.6666 


42 


29.1666 


12 


.2222 


42 


2.7222 


13 


7.2824 


43 


30.0601 


13 


.2608 


43 


2.8646 


14 


7.9074 


44 


30.9629 


14 


.3025 


44 


2.9876 


16 


8.6416 


46 


31.8750 


16 


.3472 


46 


3.1250 


16 


9.1861 


46 


32.7962 


16 


.3951 


46 


3.2654 


17 


9.8379 


47 


33.7268 


17 


.4460 


47 


3.4089 


18 


10.6000 


48 


34.6666 


18 


.5000 


48 


3.6565 


19 


11.1712 


49 


36.6167 


19 


.6571 


49 


3.7052 


20 11.8618 


60 


36.6740 


20 


.6173 


60 


3.8580 


21 


12.6416 


61 


37.6416 


21 


.6805 


61 


4.0189 


22 


13.2407 


62 


38.6186 


22 


.7469 


52 


41728 


23 


13.9400 


63 


39.6046 


23 


.8163 


63 


4.3349 


24 


14.6666 


64 


40.6000 


24 


.8889 


64 


45000 


26 


16.3936 


66 


41.6046 


26 


.9647 


56 


46682 


26 


16.1296 


66 


42.6186 


26 


1.0432 


56 


48396 


27 


16.8760 


67 


43.6416 


27 


1.1250 


67 


6.0139 


28 


17.6296 


68 


44.6740 


28 


1.2099 


58 


6.1913 


29 


18.3936 


69 


46.6167 


29 


1.2978 69 


6.3719 


30 


19.1666 


60 1 46.6666 


30 


1.3889 60 


5.5666 







( Ixxii. ) 






BASE 27— SLOPE i to 1. 


1 


Add. 


1 

•55 


Add. 


if. of 
Hts. 


Deduct. 


Dif.of 
Hts. 


Deduct. 


a 




a: 




O 






1 


.6092 


31 


24.3981 


1 


.0031 


31 


2.9660 


2 


1.0370 


32 


26.4814 


2 


.0123 


32 


3.3611 


3 


1.5833 


33 


26.6833 


3 


.0278 


33 


3.6111 


4 


2.1481 


34 


27.7037 


4 


.0494 


34 


3.5679 


d 


2.7314 


36 


28.8426 


6 


.0772 


35 


3.7808 


6 


3.3333 


36 


30.0000 


6 


.1111 


36 


40000 


7 


3.9536 


37 


31.1758 


7 


.1612 


37 


42253 


8 


45926 


38 


32.3703 


8 


,1976 


38 


44568 


9 


5.2500 


39 


33.6833 


9 


,2500 


39 


46944 


10 


5.9258 


40 


34.8148 


10 


.3086 


40 


49383 


11 


6.6203 


41 


36.0648 


11 


.3734 


41 


5.1883 


12 


7.3383 


42 


37.3333 


12 


.4444 


42 


5.4444 


13 


8.0648 


43 


38.6203 


13 


.6216 


43 


5.7076 


14 


8.8148 


44 


39.9268 


14 


.6049 


44 


5.9753 


15 


9.5833 


45 


41.2500 


16 


.6944 


45 


6.2500 


16 


10.3703 


46 


42.5926 


16 


.7901 


46 


&6308 


17 


11.1758 


47 


43.9536 


17 


.8920 


47 


6.8179 


18 


12.0000 


48 


45.3333 


18 


1.0000 


48 


7.1111 


19 


12.8425 


49 


46.7314 


19 


1.1142 


49 


7.4105 


20 


13.7037 


60 


48.1481 


20 


1.2346 


50 


7.7160 


21 


14.5833 


61 


49.5833 


21 


1.3611 


51 


8.02T7 


22 


15.4814 


52 


61.0370 


22 


1.4938 


62 


8.3466 


23 


16.3981 


53 


52.6092 


23 


1.6327 


63 


8.6607 


24 


17.3333 


54 


540000 


24 


1.7778 


64 


9.0000 


25 


18.2870 


65 


55,6092 


26 


1.9296 


66 


9.3364 


26 


19.2692 


56 


67.0370 


26 


2.0864 


56 


9.6790 


27 


20.2500 


67 


68.5833 


27 


2.2500 


57 


10.0277 


28 


21.2592 


68 


60.1481 


28 


2.4197 


58 


10.3827 


29 


22.2870 


69 


61.7314 


29 


2.6956 


59 


10.7438 


30 


23.3333 


60 


63.3333 


30 


2.7778 


60 


11.1111 



( Izxiii. ) 
BASE 27— SLOPE i to 1. 



•s 


Add. 




Add. 




Deduct. 


3^ 


Deduct. 


X 




K 




Q 




o 




1 


.5138 


31 


28.8470 


1 


.0046 


31 


4.4490 


2 


1.0555 


32 


30.2222 


2 


.0185 


32 


4.7407 


3 


1.6250 


33 


31.6250 


3 


.0416 


33 


5.0416 


4 


2.2222 


34 


33.0565 


4 


.0740 


34 


5.3518 


6 


2.8472 


35 


34.5138 


5 


.1167 


36 


5.6712 


6 


3.5000 


36 


36.0000 


6 


.1667 


36 


6.0000 


7 


4.1804 


37 


37.6138 


7 


.2268 


37 


6.3379 


8 


4.8888 


38 


39.0665 


8 


.2963 


38 


&6851 


9 


5.6250 


39 


40.6250 


9 


.3750 


39 


7.0416 


10 


6.3888 


40 


42.2222 


10 


.4630 


40 


7.4074 


11 


7.1805 


41 


43.8472 


11 


.5602 


41 


7.7824 


12 


8.0000 


42 


45.5000 


12 


.6667 


42 


8.1666 


13 


8.8472 


43 


47.1805 


13 


.7824 


43 


8.5612 


14 


9.7222 


44 


48.8888 


14 


.9074 


44 


8.9629 


16 


10.6250 


45 


so.eoso 


16 


1.0417 


45 


9.3750 


16 


11.5655 


46 


52.3888 


16 


1.1862 


46 


9.7962 


17 


12.5138 


47 


64.1805 


17 


1.3379 


47 


10.2268 


18 


13.5000 


48 


56.0000 


18 


1.5000 


48 


10.6666 


19 


14.5138 


49 


57.8472 


19 


1.6713 


49 


11.1157 


20 


15.6555 


50 


69.7222 


20 


1.8618 


50 


11.5740 


21 


16.6260 


51 


61.6250 


21 


2.0417 


51 


12.0416 


22 


17.7222 


52 


63.5555 


22 


2.2407 


52 


12.5186 


23 


18.8472 


53 


66.6138 


23 


2.4491 


53 


13.0046 


24 


20.0000 


54 


67.6000 


24 


2.6667 


54 


13.5000 


25 


21.1806 


55 


69.5138 


26 


2.8935 


55 


14.0046 


26 


22.3888 


66 


71.5566 


26 


3.1296 


66 


14.5185 


27 


23.6250 


57 


73.6260 


27 


3.8760 


67 


15.0416 


28 


24.8888 


58 


75.7222 


28 


3.6296 


58 


16.6740 


29 


26.1804 


59 


77.8472 


29 


a8985 


66 


16.1157 


30 


27.5000 


60 


80.0000 


30 


4.1667 


60 


16.6666 





( Izxiv. ) 




BASE 27— SLOPE 1 to 1. 


t 


Add. 


1 

n 


A<M. 


Dif.of 
Hti. 


Deduct. 


Dif.of 
Hto. 


Deduct. 


1 


.5184 


31 


33.2962 


1 


.0062 


31 


5.9321 


3 


1,0740 


82 


84.9629 


2 


.0247 


32 


&3210 


3 


1 £iti£i£t 
l.OdQD 


33 


86.6666 


8 


.0556 


33 


ft7222 


4 


2.2962 


84 


38.4074 


4 


.0988 


34 


7.1368 


5 


2.9629 


35 


40.1861 


5 


.1543 


36 


7.6617 


6 




36 


42.0000 


6 


.2222 


36 


8.0000 


7 


44074 


87 


43.8618 


7 


.3025 


37 


8.4506 


8 


5.1851 


88 


45.7407 


8 


.3951 


38 


8.9135 


9 


&0000 


39 


47.6666 


9 


.5000 


39 


9.3888 


10 


6.8518 


40 


49.6296 


10 


.6173 


40 


9.8765 


11 


7.7407 


41 


51.6296 


11 


.7469 


41 


10.3765 


12 


8.6666 


42 


53.6666 


12 


.8889 


42 


10.88iS8 


13 


9.6296 


43 


68.7407 


13 


1.0432 


43 


11.4135 


14 


10.6296 


44 


67.8618 


14 


1.2099 


44 


11.9506 


16 


11.6666 


45 


60.0000 


15 


1.3889 


46 


13.6000 


16 


12.7407 


46 


62.1851 


16 


1.5802 


46 


13.0617 


17 


13.8518 


47 


64.4074 


17 


1.7839 


47 


13.6358 


18 


15.0000 


48 


66.6666 


18 


2.0000 


48 


142222 


19 


16.1861 


49 


68.9629 


19 


2.2284 


49 


148209 


20 


17.4074 


50 


71.3962 


20 


2.4691 


60 


15.4^1 


21 


18.6666 


51 


73.6666 


21 


2.7222 


61 


16.0555 


22 


19.9629 


m 


76.0740 


22 


2.9876 


52 


lft6913 


23 


21.2962 


53 


78.6184 


23 


3.2664 


63 


17.3395 


24 


22.6666 


54 


81.0000 


24 


3.6655 


54 


18.0000 


25 


34.0740 


65 


83.5184 


26 


3.8630 


66 


18.6728 


26 


25.5184 


66 


86.0740 


26 


41728 


56 


19.3680 


27 


37,0000 


67 


88.6666 


27 


46000 


67 


20.0556 


28 


28.6184 


68 


91.2962 


28 


48395 


68 


20.7654 


39 


30.0740 


59 


93.9629 


29 


6.1913 


59 


21.4876 


30 


31.6666 


60 


96.6666 

• 


30 


6.5666 


60 


22.2222 



(IXIT. ) 




BASE 27— SLOPE 


1| to 1. 


t 


Add. 


1 


• 

Add. 


if. of 

Ht8. 


Dedoet. 


3^ 


Deduct. 


» 




» 




Q 




Q 




1 


.5230 


31 


37.7453 


1 


.0077 


31 


7.4161 


2 


1.0926 


32 


39.7036 


2 


.0809 


32 


7.9012 


3 


1.7083 


33 


41.7083 


3 


.0694 


33 


8.4027 


4 


2.3703 


34 


43.7692 


4 


.1234 


34 


8.9197 


5 


3.0788 


35 


45.8564 


5 


.1928 


86 


9.4521 


6 


3.8333 


36 


48.0000 


6 


.2778 


36 


10.0000 


7 


4.6342 


37 


50.1897 


7 


.3781 


37 


10.6632 


8 


5.4814 


38 


62.4259 


8 


.'^38 


38 


11.1419 


9 


6.3750 


39 


64.7083 


9 


.6250 


39 


11.7361 


10 


7.3148 


40 


57.0870 


10 


.7716 


40 


12.3456 


11 


8.3009 


41 


69.4120 


11 


.9386 


41 


12.9706 


12 


9.3^3 


42 


61.8833 


12 


1.1111 


42 


18.6111 


13 


10.4120 


43 


64.3009 


13 


1.3040 


43 


14.2680 


14 


11.5370 


44 


66.8147 


14 


1.6123 


44 


14.9382 


15 


12.7082 


45 


69.3760 


16 


1.7361 


45 


15.6250 


16 


13.9259 


46 


71.9814 


16 


1.97SS 


46 


16J3271 


17 


15.1897 


47 


74.6342 


17 


2.2299 


47 


17/)447 


18 


16.6000 


48 


77.8338 


18 


2.5000 


48 


17.7777 


19 


17.8564 


49 


80.0788 


19 


2.7856 


49 


18JiQ6& 


20 


19.2592 


50 


82.8708 


20 


8.0864 


60 


1942901 


21 


20.7083 


51 


86.7088 


21 


8.4028 


61 


20.0694 


22 


22.2036 


52 


88.5926 


2J 


3.7346 


BH 


20.8641 


23 


23.7463 


53 


91.5230 , 


23 


4.0818 


53 


21.6743 


24 


25.3333 


54 


94.5000 


24 


4.4444 


54 


22.5000 


25 


26.9675 


56 


97.5230 


26 


4.8228 


m 


23^10 


26 


26.6480 


56 


100.6926 


26 


5.2160 


56 


24.1976 


27 


30.3760 


67 


103.7083 


27 


5.6250 


57 


25.0694 


28 


32.1481 


58 


106.8708 


28 


6.0^^4 


68 


25.9567 


29 


33.9675 


59 


110.0788 


29 


6.4891 


59 


26.8596 


30 


35.8333 


60 


113.8333 


30 


6.9444 


60 


27.7777 



( Ixxvi. ) 
BASE 27— SLOPE IJ to 1. 



33 



1 

2 
3 
4 
5 

6 
7 
8 
9 
10 

11 
12 
13 
14 
15 

16 
17 
18 
19 
20 

21 
22 
23 
24 
25 

26 
27 
28 
^^ 
30 



Add. 



.5277 
1.1111 
1.7500 
2.4444 
3.1944 

4.0000 
4.8610 
5.7777 
6.7500 

7.7777 

8.8610 
10.0000 
11.1944 
12.4444 
13.7500 

15.1111 
16.5277 
18.0000 
19.5277 
21.1111 

22.7500 
24.4444 
26.1944 
28.0000 
29.8610 

31.7777 
33.7500 
35.7777 
37.8610 
40.0000 



-a 

•s 



31 
32 
33 
34 
35 

36 
37 
38 
39 
40 

41 
42 
43 
44 
45 

46 
47 
48 
49 
50 

51 
52 
53 
54 
55 

56 
57 
58 
59 
60 



Add. 



42.1944 
44.4444 
46.7500 
49.1111 
51.5277 

54.0000 
56.5277 
59.1111 
61.7500 
64.4444 

67.1944 
70.0000 
72.8610 
75.7777 
78.7500 

81.7777 
84.8610 
88.0000 
91.1944 
94.4444 

97.7500 
101.1111 
104.5277 
108.0000 
111.5277 

115.1111 
118.7500 
122.4444 
126.1944 
130.0000 






1 

2 
3 
4 
5 

6 

7 

8 

9 

10 

11 
12 
13 
14 
15 

16 
17 
18 
19 
20 

21 
22 
23 
24 
25 

26 
27 
28 
29 
30 



Deduct. 



.0092 
.0370 
.0833 
,1480 
.2313 

.3333 
.4537 
.5926 
.7500 
.9259 

1.1203 
1.3333 
1.5648 
1.8148 
2.0833 

2.3704 
2.6759 
3.0000 
3.3426 
3.7037 

4.0833 
4.4815 
4.8981 
5.3333 
5.7869 

6.2592 
6.7500 
7.2592 
7.7869 
8.3333 



^5 



s 



Deduct. 



31 8.8981 

32 9.4815 

33 10.0833 

34 10.7037 

35 11.3425 

36 12.0000 

37 12.6759 

38 13.3703 

39 14.0833 

40 14.8148 

41 15.5648 

42 16.3333 

43 17.1202 

44 17.9259 

45 18.7500 

46 19.5925 

47 20.4537 

48 21.3333 

49 22.2314 

50 23.1481 

51 24.0833 

52 25.0370 

53 26.0092 

54 27.0000 

55 28.0092 

56 29.0370 

57 30.0833 

58 31.1481 

59 32.2314 

60 33.3333 



• 


( Ixxvii. ) 




BASE 27— SLOPE 


1* to 1. 


1 

» 


Add. 


i 


Add. 


1 


Deduct. 


Dif. of 
Hts. 


Deduct. 


1 


.5323 


31 


46.6435 


.0108 


31 


10.3811 


2 


1.1296 


32 


49.1851 


2 


.0432 


32 


11.0617 


3 


1.7916 


33 


51.7916 


3 


.0972 


33 


11.7638 


4 


2.5185 


34 


54.4629 


4 


.1728 


34 


12.4876 


5 


3.3101 


35 


67.1990 


5 


.2700 


36 


13.2330 


6 


4.1666 


36 


60.0000 


6 


.3889 


36 


14.0000 


7 


5.0878 


37 


62.8636 


7 


.5293 


37 


14.7886 


8 


6.0740 


38 


66.7962 


8 


.6913 


38 


16.6987 


9 


7.1250 


39 


68.7916 


9 


.8760 


39 


16.4305 


10 


8.2406 


40 


71.8518 


10 


1.0802 


40 


17.2839 


11 


9.4213 


41 


74.9768 


11 


1.3071 


41 


18.1689 


12 


10.6666 


42 


78.1666 


12 


1.6566 


42 


19.0556 


13 


11.9768 


43 


81.4212 


13 


1.8256 


43 


19.9747 


14 


13.3518 


44 


84.7407 


14 


2.1173 


44 


20.9136 


16 


147916 


45 


88.1260 


15 


2.4305 


45 


21.8750 


16 


16.2962 


46 


91.6740 


16 


2.7654 


46 


22.8580 


17 


17.8666 


47 


95.0878 


17 


3.1219 


47 


23.8326 


18 


19.6000 


48 


98.6666 


18 


3.5000 


48 


24.8888 


19 


21.1990 


^ 


102.3101 


19 


3.8997 


49 


25.9367 


20 


22.9629 


50 


106.0185 


20 


4.3210 


50 


27.0061 


21 


24.7916 


51 


109.7916 


21 


4.7639 


51 


28.0972 


22 


26.6851 


52 


113.6296 


22 


6.2284 


52 


29.2098 


23 


28.6436 


63 


117.5323 


23 


5.7145 


53 


30.3441 


24 


30.6666 


54 


121.5000 


24 


6.2222 


54 


31.6000 


23 


32.7645 


55 


126.5323 


26 


6.7616 


55 


32.6774 


26 


34.9073 


56 


129.6296 


26 


7.3024 


56 


33.8765 


27 


37.1250 


67 


133.7916 


27 


7.8750 


57 


35.0972 


28 


39.4073 


68 


138.0185 


28 


8.4691 


68 


36.3394 


29 


41.7545 


69 


142.3101 


29 


9.0848 


69 


37.603& 


30 


44.1666 


60 


146.6666 


30 


9.7222 


60 


38.8888 





( IxxTiii. ) 


• 


BASE 27-8LOPE 2 to 1. 


i-> Height 


Add. 


1 

s 


Add. 


1 


Deduct 


Dif.of 

HtB. 


Deduct. 


.6370 


31 


51.0926 


.0123 


31 


11.8642 


2 


1.1481 


32 


63.9268 


2 


.0494 


32 


12.6419 


3 


1.8333 


33 


56.8333 


3 


.1111 


33 


13.4444 


4 


2.5926 


34 


59.8147 


4 


.1975 


34 


142716 


6 


3.4258 


35 


62.8703 


6 


.3086 


35 


16.1234 


6 


4.3333 


36 


66.0000 


6 


.4444 


36 


16.0000 


7 


5.3148 


37 


69.2037 


7 


.6049 


37 


16.9012 


8 


6.3703 


38 


72.4814 


8 


.7901 


38 


17.8271 


9 


7.6000 


39 


76.8333 


9 


1.0000 


39 


18.7777 


10 


8.7037 


40 


79.2592 


10 


1.2346 


40 


19.7530 


11 


9.9816 


41 


82.7692 


11 


1.4938 


41 


20.7530 


12 


11.3333 


42 


86.3333 


12 


1.7778 


42 


21.7777 


13 


12.7692 


43 


89.9814 


13 


2.0864 


43 


22.8271 


14 


14.2692 


44 


93.7037 


14 


2.4197 


44 


23.9012 


15 


16.8333 


45 


97.6000 


15 


2.7778 


45 


25.0000 


16 


17.4815 


46 


101.3708 


16 


3.1605 


46 


26.1234 


17 


19.2037 


47 


105.3147 


17 


3.5679 


47 


27.2716 


18 


21.0000 


48 


109.3333 


18 


40000 


48 


28.4444 


19 


22.8703 


49 


113.4258 


19 


44568 


49 


29.6420 


20 


248148 


60 


117.6926 


20 


49382 


50 


30.8642 


21 


26.8333 


51 


121.8333 


21 


5.4444 


51 


32.1111 


22 


289258 


62 


126.1480 


22 


5.9753 


62 


33.3827 


23 


31.0926 


63 


130^870 


23 


6.6309 


53 


346790 


24 


33.3333 


54 


136.0000 


24 


7.1111 


64 


36.0000 


25 


36.6480 


66 


139.6370 


25 


7.7160 


66 


37.3456 


26 


38.0370 


66 


144.1480 


26 


8.3457 


56 


38.7160 


27 


40.6000 


67 


148.8333 


27 


9.0000 


57 


40.1111 


28 


43.0370 


68 


153.5926 


28 


9.6790 


68 


41.6308 


29 


46.6480 


59 


158.4268 


29 


10.3827 


69 


42.9763 


SO 


48.3333 


60 


163.3333 


30 


11.1111 


60 


444444 







( Izxix. ) 




BASE 27 SLOPE 2| to 1. 


1 


Add. 


1 

n 


Add. 


Dif. of 

HtB. 


Deduct. 


Dif. of 
Hto. 


Deduct. 


1 


.5462 


31 


59.9908 


1 


.0154 


31 


148302 


2 


1.1851 


32 


63.4073 


2 


.0617 


32 


15.8013 


3 


1.9166 


33 


66.9166 


3 


.1389 


33 


1&8066 


4 


2.7408 


34 


70.5186 


4 


.2468 


34 


17.8895 


6 


3.6573 


35 


742129 


5 


.3857 


35 


18.9043 


6 


4.6666 


36 


78.0000 


6 


.5556 


36 


20.0000 


7 


5.7686 


37 


81.8796 


7 


.7661 


37 


21.1266 


8 


6.9629 


38 


86.8619 


8 


.9876 


38 


22.2839 


9 


8.2600 


39 


89.9167 


9 


1.2500 


39 


23.4722 


10 


9.6296 


40 


940740 


10 


1.5432 


40 


246913 


11 


11.1018 


41 


98.3240 


11 


1.8673 


41 


26.9413 


12 


12.6666 


^ 


102.6666 


12 


2.2222 


42 


27.2222 


13 


14.3240 


43 


107.1018 


13 


2.6080 


43 


28.5360 


14 


ia0740 


44 


111.6296 


14 


3.0247 


44 


29.8765 


15 


17.9167 


45 


116.2500 


15 


3.4722 


46 


31.2600 


16 


19.8519 


46 


120.9629 


16 


3.9606 


46 


32.6643 


17 


21.8796 


47 


125.7686 


17 


44599 


47 


340896 


18 


240000 


48 


130.6666 


18 


5.0000 


48 


36.5666 


19 


26.2120 


49 


135.6573 


19 


5.5710 


49 


37.0624 


20 


28.5185 


50 


140.7408 


20 


6.1728 


50 


38.6802 


21 


30.9166 


51 


145.9166 


21 


6.8066 


51 


40.1388 


22 


33.4073 


82 


161.1851 


22 


7.4691 


62 


41.7283 


23 


35.9906 


53 


156.5462 


23 


8.1636 


53 


43.3487 


24 


38.6666 


54 


im.(mo 


24 


8.8889 


54 


45.0000 


26 


^4352 


55 


167.5462 


26 


9.6466 


56 


46.6820 


26 


4420^ 


56 


173.1851 


26 


10.4*21 


66 


48.3960 


27 


47.2500 


67 


178.9166 


27 


11.2500 


57 


60.1388 


28 


50.2962 


58 


184.7408 


28 


12.0687 


68 


61.9135 


29 


53.4352 


59 


190.6573 


29 


12.9782 


59 


63.7191 


30 

.: 


5&6666 


60 


196.6666 


30 


13.8889 


60 


66.6665 



" 












( Ixxx. ) 






BASE 27— SLOPE 3 to 1. 




> 


i 

a 
1 


Add. 


} 


Add. 


R 

1 


Dedoct. 


Dif.of 
Hts. 


Deduct. 




.5655 


31 


68.8888 


.0185 


31 


17.7963 




2 


1.2222 


32 


72.8888 


2 


.0740 


32 


18.9630 




3 


2.0000 


33 


77.0000 


3 


.1667 


33 


20.1666 




4 


2.8888 


34 


81.2222 


4 


.2963 


34 


21.4074 




6 


3.8888 


35 


86.5666 


5 


.4628 


36 


22.6851 




6 


5.0000 


36 


90.0000 


6 


.6667 


36 


24.0000 




7 


6.2222 


37 


94.5565 


7 


.9074 


37 


25.3618 




8 


7.5556 


38 


99.2222 


8 


1.1862 


38 


26.7407 




9 


9.0000 


39 


104.0000 


9 


1.5000 


39 


28.1666 




10 


10.6565 


40 


108.8888 


10 


1.8518 


40 


29.6296 




11 


12.2222 


41 


113.8888 


11 


2.2407 


41 


31.1296 




12 


14.0000 


42 


119.0000 


12 


2.6667 


42 


32.6666 




13 


16.8888 


43 


124.2222 


13 


3.1296 


43 


34.2407 




14 


17.8888 


44 


129.6566 


14 


3.6296 


44 


36.8518 




15 


20.0000 


46 


135.0000 


15 


4.1667 


46 


37.6000 




16 


22.2222 


46 


140.5665 


16 


4.7407 


46 


39.1851 




17 


24.5655 


47 


146.25^22 


17 


6.3618 


47 


40.9074 




18 


27.0000 


48 


152.0000 


18 


6.0000 


48 


42.6666 




19 


29.6556 


49 


157.8888 


19 


6.6862 


49 


44.4629 




20 


32.JK222 


50 


163.8888 


20 


7.4074 


50 


46.2963 




21 


35.0000 


51 


170.0000 


21 


8.1667 


51 


49.1666 




22 


37.8888 


52 


176.2222 


22 


8.9629 


62 


60.0740 




■ 23 


40.8888 


53 


1 82.5555 


23 


9.7962 


53 


52.0186 




24 


44.0000 


54 


189.0000 


24 


10.6667 


54 


64.0000 




26 


48.2222 


55 


196.5555 


25 


11.5741 


66 


56.0184 




26 


60.5556 


56 


202.2222 


26 


12.5184 


56 


58.0740 




27 


64.0000 


67 


209.0000 


27 


13.6000 


57 


60.1666 




28 


67.5556 


68 


215.8888 


28 


14.5185 


58 


62.2962 




29 


61.2222 


59 


222.8888 


29 


15.5741 


59 


64.4629 


1 


30 


65.0000 


60 


230.0000 


30 


16.6667 


60 


66.6666 





(Ixxxi. ) 


W|BA8E 28— SLOPE i to 1. 


1 


Add. 


1 


Add. 




Deduct. 




Deduct. 


» 




IS 




1 




O* 




1 


.5231 


31 


20.6231 


.0015 


81 


1.4830 


2 


1.0555 


32 


21.3333 


2 


.0062 


32 


1.6802 


3 


1.5972 


33 


22.1627 


8 


.0189 


33 


1.6805 


4 


2.1481 


34 


22.9814 


4 


.0246 


34 


1.7839 


5 


2.7083 


35 


23.8194 


5 


.0386 


35 


1.8904 


6 


3.2777 


36 


24.6666 


6 


.0555 


36 


2.0000 


7 


3.8564 


37 


25.6231 


7 


.0756 


37 


2.1126 


8 


4.4444 


38 


26.3888 


8 


.0988 


38 


2.2284 


9 


5.0416 


39 


27.2639 


9 


.1250 


39 


2.8472 


10 


5.6481 


40 


28.1481 


10 


.1543 


40 


2.4691 


11 


6.2639 


41 


29.0416 


11 


.1867 


41 


2.5941 


12 


6.8888 


42 


29.9444 


12 


.2222 


42 


2.7222 


13 


7.5231 


43 


30.8564 


13 


.2608 


43 


2.8545 


14 


8.1666 


44 


31.7777 


14 


.3025 


44 


2.9876 


15 


8.8194 


45 


32.7083 


16 


.8472 


45 


3.1250 


16 


9.4814 


46 


33.6481 


16 


.3951 


46 


3.2654 


17 


10.1627 


47 


34.6972 


17 


.4460 


47 


8.4089 


18 


10.8333 


48 


35.5555 


18 


.6000 


48 


3.6555 


19 


11.6231 


49 


36.5231 


19 


.6571 


49 


3.7052 


20 


12.2222 


50 


37.5000 


20 


.6173 


50 


3.8580 


21 


12.9305 


61 


38.4861 


21 


.6806 


51 


4.0139 


22 


13.6481 


52 


39.4814 


22 


.7469 


52 


4.1728 


23 


143750 


53 


40.4861 


23 


.8163 


53 


4.3349 


24 


15.1111 


64 


41.5000 


24 


.8889 


64 


4.6000 


25 


15.8664 


56 


42.6231 


26 

• 


.9647 


65 


46682 


26 


16.6111 


66 


43.6666 


26 


1.0432 


66 


48395 


27 


17.3760 


57 


44.5972 


27 


1.1250 


57 


5.0139 


28 


18.1481 


58 


46.6481 


28 


1.2099 


68 


6.1913 


29 


18.9305 


69 


46.7083 


29 


1.2978 


69 


5.8719 


30 


19.7222 


60 


47.7777 


30 


1.8889 60 

H 


5.5566 

■ 





( Ixxxii. ) 


BASE 28— SLOPE ^ to 1. 


5 




S 




O • 








1 


Add. 


•r 

31 


Add. 




Deduct. 


gn 


Deduct. 


.6277 


249722 


1 


.0031 


31 


2.9660 


2 


1.0740 


32 


26.0740 


2 


.0123 


32 


3.1606 


3 


1.6368 


33 


27.1944 


3 


.0278 


33 


3.3611 


4 


2.2222 


34 


28.3333 


4 


.0494 


84 


3.5679 


6 


2.8240 


35 


29.4907 


6 


.0772 


35 


3.7808 


6 


a4444 


36 


30.6666 


6 


.1111 


86 


4.0000 


7 


40833 


37 


31.8611 


7 


.1512 


37 


4.2263 


8 


47407 


38 


33.0740 


8 


.1976 


38 


4.4568 


9 


5.4166 


39 


343056 


9 


.2600 


39 


4.6944 


10 


anil 


40 


35.5555 


10 


•oOoD 


40 


49383 


11 


&8240 


41 


36.8240 


11 


.3734 


41 


5.1883 


12 


7.5655 


^ 


38.1111 


12 


.4444 


42 


5.4444 


13 


8.3055 


43 


39.4166 


13 


.5216 


43 


6.7067 


14 


9.0740 


44 


40.7407 


14 


.6049 


44 


6.9763 


15 


9.8611 


46 


^.0633 


15 


.6944 


45 


&2500 


16 


10.6666 


46 


43.4444 


16 


.7901 


46 


6.5308 


17 


11.^06 


47 


44.8240 


17 


.8920 


47 


6.8179 


18 


12.3333 


48 


46i2222 


18 


1.0000 


48 


7.1111 


19 


13.1944 


49 


47.6889 


19 


1.11<^ 


49 


7.4106 


20 


14.0740 


50 


49.0740 


20 


1.2346 


50 


7.7160 


21 


149722 


51 


50.62T7 


21 


1.3611 


51 


8.0277 


22 


16.8888 


52 


52.0000 


22 


1.4938 


52 


8.3466 


23 


1&8240 


53 


53.^07 


23 


1.6327 


53 


8.6697 


24 


17.7777 


54 


55.0000 


24 


1.7778 


54 


9.0000 


25 


18.7600 


55 


56.6277 


25 


1.9295 


55 


9.3364 


26 


19.7407 


56 


58.0740 


26 


2.0864 


66 


9.6790 


27 


20.7600 


57 


69.6389 


27 


2.2600 


57 


10.0277 


28 


21.7777 


68 


61.229ia 


28 


2.4197 


58 


10.3827 


29 


22.8240 


69 


^.8240 


29 


2.5956 


59 


10.7438 


30 


23.8888 


60 


64.4444 


30 


2.7778 


60 


11.1111 





( Ixxziii. ) 


BASE 28 SLOPE f to 1. 


1 

a 


Add. 


1 

s 

31 


Add. 


1 


Deduct. 




Deduct. 


1 


.5323 


29.4218 


.0046 


31 


4.4490 


2 


L0925 


32 


30.8148 


2 


.0185 


32 


47407 


8 


1.6805 


33 


32.2861 


8 


.0416 


33 


5.0416 . 


4 


2.29^ 


84 


33.6851 


4 


.0740 


34 


5.3518 


6 


2.9397 


35 


36.1620 


5 


.1157 


36 


66712 


6 


3.6111 


36 


36.6666 


6 


.1667 


86 


6.0000 


7 


4.3101 


37 


38.1990 


7 


.2268 


37 


6.8379 


8 


6.0370 


38 


39.7692 


8 


.2963 


38 


6.6851 


9 


6.7916 


39 


41.3470 


9 


.3750 


89 


7.0416 


10 


6.6740 


40 


42.9629 


10 


.4630 


40 


7.4074 


11 


7.8842 


41 


44.6064 


11 


.6602 


41 


7.7824 


12 


8.2222 


42 


46.2777 


12 


.6667 


42 


8.1666 


13 


9.0879 


43 


47.9768 


13 


.7824 


48 


8.5602 


14 


9.9815 


44 


49.7037 


14 


.9074 


44 


8.9629 


15 


10.9027 


46 


51.4583 


15 


1.0417 


46 


9.3760 


16 


11.8518 


46 


63.2407 


16 


1.18S2 


46 


9.7962 


17 


12.8286 


47 


55.0508 


17 


1.3379 


47 


10.2268 


18 


1^8333 


48 


56.8888 


18 


1.6000 


48 


10.6666 


19 


14.8657 


49 


58.7647 


19 


1.6713 


49 


11.1167 


20 


15.9258 


60 


60.6481 


20 


1.8618 


50 


11.5740 


i 21 


17.0138 


51 


62.6694 


21 


2.0417 


51 


12.0416 


22 


18.1296 


52 


64.6185 


22 


2.2407 


62 


12.6185 


23 


19.2731 


53 


66.4953 


23 


2.4491 


53 


13.0046 


24 


20.4444 


54 


68.6000 


24 


2.6667 


54 


13.6000 


25 


21.6435 


65 


70.6328 


26 


2.8935 


55 


14.0046 


26 


22.8703 


56 


72.6926 


26 


3.1296 


56 


14.6185 


27 


24.1250 


67 


74.6806 


27 


3.8750 


57 


15.0416 


28 


25.4074 


68 


76.7962 


28 


8.6296 


58 


15.5740 


29 


26.7176 


69 


78.9397 


29 


8.8936 


59 


16.1157 


30 


28.0556 


60 


81.1111 


80 4.1667 


60 


16.6666 





( Ixxxiv. ) 


BASE 28— SLOPE 1 to 1. 1 


1 


Add. 




Add. 


Dif.of 
Hte. 


Deduct. 


Dif.of 
Hit. 


Deduct* 


1 


U5870 


81 


83.8704 


I 


.0062 


31 


5.9321 


2 


1.1111 


82 


35.6555 


2 


.0247 


82 


6.3210 


3 


1.7222 


38 


87.2777 


8 


.0555 


38 


6.7222 


4 


2.370S 


84 


39.0370 


4 


.0988 


84 


7.1358 


6 


3.0555 


35 


40.8333 


5 


.1543 


85 


7.5617 


6 


8.7777 


86 


42.6666 


6 


.2222 


36 


8.0000 


7 


4.5370 


87 


445370 


7 


.3025 


37 


8.4506 


8 


5.3333 


88 


46.4444 


8 


.3951 


38 


8.9135 


9 


6.1666 


89 


48.3888 


9 


.5000 


89 


9.3888 


10 


7.0370 


40 


50.3703 


10 


.6173 


40 


9.8765 


n 


7.9444 


41 


52.8888 


11 


.7469 


41 


10.3765 


12 


8.8888 


4Si 


54.4444 


12 


.8889 


42 


10.8888 


13 


9.8708 


43 


56.5370 


13 


1.0432 


43 


11.4135 


14 


10.8888 


44 


68.6666 


14 


1.2099 


44 


11.9506 


15 


11.9444 


45 


60.8333 


16 


1.3889 


46 


12.5000 


16 


13.0370 


46 


63.0370 


16 


1.6802 


46 


13.0617 


17 


14.1666 


47 


65.2777 


17 


1.7839 


47 


13.6358 


18 


15.3333 


48 


67.6566 


18 


2.0000 


48 


14.2222 


19 


16.6370 


49 


69.8704 


19 


2.2284 


49 


14.8209 


20 


17.7777 


50 


72.2222 


20 


2.4691 


50 


15.4321 


21 


19.0555 


51 


746111 


21 


2.7222 


51 


16.0555 


22 


20.3703 


52 


77.0370 


22 


2.9876 


52 


16.6913 


23 


21.7222 


53 


79.6000 


23 


3.2654 


53 


17.3395 


24 


23.1111 


54 


82.0000 


24 


3.5655 


54 


18.0000 


25 


24.5370 


55 


84,6370 


25 


3.8680 


55 


18.6728 


26 


26.0000 


56 


86.1111 


26 


41728 


56 


19.3580 


27 


27.6000 


57 


89.7222 


27 


46000 


57 


20.0555 


28 


29.0370 


58 


92.3703 


28 


48396 


58 


20.7654 


29 


30.6111 


59 


95.0666 


29 


6.1913 


69 


21.4876 


30 


32.2222 60 


97.7777 


30 


5.5665 


60 


22.2222 



(Ixxxv.) 


BASE 28— SLOPE 1^ to 1. 


1 


Add. 


1 


Add. 


Dif.or 
Hts. 


Deduct. 


I>if.of 
Htf. 


Deduct. 


1 


.6416 


81 


38.3194 


1 


.0077 


81 


7.4151 


2 


1.1296 


32 


40.2962 


2 


.0309 


32 


7.9012 


3 


1.7638 


33 


42.3194 


8 


.0694 


83 


8.4027 


4 


2.4444 


34 


443888 


4 


.1234 


34 


8.9197 


5 


3.1712 


36 


46.6046 


6 


.1928 


36 


9.4621 


6 


8.9444 


36 


48.6666 


6 


.2778 


36 


10.0000 


7 


4.7638 


87 


60.8760 


7 


.3781 


37 


10.6632 


8 


6.6296 


38 


63.1296 


8 


.4938 


38 


11.1419 


9 


6.6416 


39 


66.4306 


9 


.6250 


39 


11.7361 


10 


7.6000 


40 


67.7777 


10 


.7716 


40 


12.3466 


11 


8i)046 


41 


60.1712 


11 


.9336 


41 


12.9706 


12 


9.6666 


42 


62.6111 


12 


1.1111 


42 


13.6111 


13 


10.6627 


43 


65.0972 


13 


1.3040 


48 


14.2680 


14 


11.7962 


44 


67.6296 


14 


1.6123 


44 


14.9382 


16 


12.9860 


46 


70.2088 


16 


1.7361 


46 


16.6250 


16 


14.^22 


46 


72.8333 


16 


1.9763 


46 


16.3271 


17 


16.6046 


47 


76.6046 


17 


2.2299 


47 


17.0447 


18 


16.8333 


48 


78.2222 


18 


2.5000 


48 


17.7777 


19 


18.2083 


49 


80.9860 


19 


2.7855 


49 


18.5262 


20 


19.6296 


50 


83.7962 


20 


3.0864 


50 


19.2901 


21 


21.0972 


61 


86.6627 


21 


3.4028 


61 


20.0694 


22 


22.6111 


62 


89.6656 


22 


3.7346 


62 


20.8641 


23 


24.1712 


63 


92.5046 


23 


4.0818 


63 


21.6748 


24 


26.7777 


64 


95.6000 


24 


44444 


54 


22.6000 


26 


27.4306 


66 


98.6416 


26 


48228 


66 


23.8410 


26 


29.1296 


66 


100.6296 


26 


6.2160 


66 


24.1976 


27 


30.8760 


67 


104.7638 


27 


6.6250 


67 


26.(J694 


28 


32.6666 


68 


107.9444 


28 


6.0494 


68 


26.9567 


28 


34.5046 


69 


111.1712 


29 


6.4891 


69 


26.8596 


30 136.3888 


60 


1144444 


30 


&9444 


60 


27.7777 

1 







( Ixxzvi. ) 




BASE 28— SLOPE 1^ to 1. 




Add. 


1 


Add. 


1 


Dednct. 




Dedaet. 


1 


.5462 


31 


42.7686 


.0092 


81 


8.8981 


2 


1.1481 


32 


45.0370 


2 


.0370 


32 


9.4815 


3 


1.8055 


33 


47.3611 


3 


.0833 


83 


10.0833 


4 


2.5185 


34 


49.7407 


4 


.1480 


34 


10.7087 


5 


3i2869 


35 


52.1759 


5 


.2813 


35 


11.3^125 


6 


4.1111 


36 


54.6666 


6 


.3388 


86 


12.0000 


7 


4.9906 


37 


57.2129 


7 


.4537 


37 


12.6769 


8 


5.9260 


38 


59.8148 


8 


.5926 


38 


13.3708 


9 


6.9166 


89 


62.4722 


9 


.7500 


39 


14.0833 


10 


7.9629 


40 


65.1851 


10 


.9259 


40 


14.8148 


11 


9.0648 


41 


67.9536 


11 


1.1203 


41 


15.5648 


12 


10.2222 


^ 


70.7777 


12 


1.3383 


42 


16.3333 


13 


11.4351 


43 


73.6574 


13 


1.5648 


48 


17.1202 - 


14 


12.7037 


44 


76.6926 


14 


1.8148 


44 


17.9259 


15 


140277 


45 


79.5833 


15 


2.0833 


46 


18.7500 


16 


15.4074 


46 


82.6296 


16 


2.3704 


46 


19.5925 


17 


16.8426 


47 


ft5.7314 


17 


2.6759 


47 


20.4637 


18 


18.3333 


48 


88.8888 


18 


3.0000 


48 


21.8338 


19 


19.8796 


49 


92.1017 


19 


SM26 


49 


22.2314 


20 


21.4816 


50 


95.3703 


20 


3.7037 


50 


23.1481 


21 


23.1388 


51 


98.6944 


21 


4.0833 


51 


24.0838 


22 


24.8518 


52 


102.0740 


22 


4.4815 


62 


26.0370 


23 


26.6203 


53 


105.5092 


23 


4.8981 


63 


26.0092 


24 


28.4444 


54 


109.0000 


24 


5.3333 


54 


27.0000 


25 


30.3240 


55 


112.5462 


25 


5.7869 


56 


28.0092 


26 


32.2592 


56 


115.1481 


26 


6.2592 


66 


29.0370 


27 


34.2500 


57 


119.8055 


27 


a7500 


57 


80.0838 


28 


36.2962 


58 


123.5185 


28 


7.2592 


68 


31.1481 


29 


38.3981 


59 


127.2870 


29 


7.7869 


59 


82.2314 


30 


40.5555 


60 


131.1111 


30 


8.8333 


60 


33.3338 





( Ixxxvii. ) 


BASE 28— SLOPE 1* to 1. 


>M Height 


Add. 


Height 


Add. 


1 


Deduct. 


Dif. of 
Uta. 


Deduct. 


.5508 


31 


47.2177 


.0108 


31 


10.3811 


2 


1.1666 


32 


49.7777 


2 


.0432 


^ 


11.0617 


3 


1.8472 


33 


62.4028 


3 


.0972 


33 


11.7638 


4 


2.5926 


34 


55.0926 


4 


.1728 


34 


12.4876 


5 


3.4027 


85 


67.8472 


5 


.2700 


36 


ia2330 


6 


4.2777 


86 


60.6666 


6 


.3889 


36 


140000 


7 


5.2177 


37 


63.6608 


7 


.5293 


37 


14.7885 


8 


6.2222 


38 


66.5000 


8 


.6913 


38 


15.6987 


9 


7.2916 


39 


69.5138 


9 


.8760 


39 


16.4305 


10 


8.4258 


40 


72.6926 


10 


1.0802 


40 


17.2839 


11 


9.6250 


41 


76.7360 


11 


1.3071 


41 


18.1689 


12 


10.8888 


4& 


78.9444 


12 


1.6656 


4Si. 


19.0566 


13 


12.2176 


48 


82.2176 


13 


1.8256 


48 


19.9747 


14 


laoiii 


44 


86.5665 


14 


2.1173 


44 


20.9185 


15 


15.0693 


45 


88.9683 


16 


2.4305 


45 


21.8760 


16 


16.5026 


46 


92.4259 


16 


2.7664 


« 


22.8580 


17 


18.1806 


47 


95.9683 


17 


3.1219 


47 


28.8626 


18 


19.8333 


48 


99.6556 


18 


a5ooo 


48 


248888 


19 


21.6609 


^ 


108.2176 


19 


3.8997 


49 


25.9867 


20 


23.3333 


60 


106.9444 


20 


4.3210 


50 


27.0061 


21 


26.1806 


51 


110.7360 


21 


4.7639 


51 


28.0972 


22 


27.0926 


52 


1146926 


22 


6.2284 


62 


29.2098 


23 


29.0693 


58 


118.6188 


23 


5.7146 


63 


80.8441 


24 


31.1111 


54 


122.6000 


24 


6.2222 


54 


81.5000 


25 


38.2176 


65 


126.5606 


25 


6.7516 


55 


32.6774 


26 


36.3888 


56 


129.6666 


26 


7.3024 


66 


33.8765 


27 


37.6250 


67 


134.8472 


27 


7.8760 


67 


86.0972 


28 


39.9258 


68 


139.0926 


28 


8.4691 


68 


36.8894 


29 


^.2916 


59 


14a4028 


29 


9.0848 


59 


87.6038 


30 


4472221 60 


147.7777 


Iso 


9.7222 \ 60 


88.8888 



( Ixxxviii. ) 


BASE 28— SLOPE 2 to 1. 


J 

n 


Add. 


t 

31 


Add. 


gw 

1 


Dednct. 


31 


Dednct. 


1 


.5555 


61.6666 


.0123 


11.8642 


2 


1.1861 


32 


54.6185 


2 


.0494 


32 


12.6419 


3 


1.8888 


33 


67.4444 


3 


.1111 


33 


13.4444 


4 


2.6666 


34 


60.4444 


4 


.1976 


34 


142716 


5 


3.5185 


35 


63.5185 


6 


.3086 


35 


16.1234 


6 


4.4444 


36 


66.6666 


6 


.4444 


36 


16.0000 


7 


5.4444 


37 


69.8888 


7 


.6049 


37 


16.9012 


8 


6.5185 


38 


73.1851 


8 


.7901 


38 


17.8271 


9 


7.6666 


39 


76.6656 


9 


1.0000 


39 


18.7777 


10 


8.8888 


40 


80.0000 


10 


1.2346 


40 


19.7530 


11 


10.1862 


41 


83.6185 


11 


l.'^38 


41 


20.7630 


12 


11.5565 


42 


87.1111 


12 


1.7778 


42 


21.7777 


13 


13.0000 


43 


90.7777 


13 


2.0664 


43 B2.8271 | 


14 


14.5185 


44 


945186 


14 


2.4197 


44 


23.9012 


15 


16.1111 


45 


98.3333 


15 


2.7778 


45 


26.0000 


16 


17.7777 


46 


102.2222 


16 


3.1605 


46 


26.1234 


17 


19.5185 


47 


106.1861 


17 


3.5679 


47 


27.2716 


18 


21.3333 


48 


110.2222 


18 


40000 


48 


28.4444 


19 


23.2222 


49 


113.3333 


19 


44668 


49 


29.6420 


20 


26.1861 


50 


118.5185 


20 


49382 


60 


30.86^ 


21 


27.2222 


51 


122.7777 


21 


6.4444 


51 


32.1111 


22 


29.3333 


52 


127.1111 


22 


6.9763 


62 


33.3827 


23 


31.5185 


53 


131.5186 


23 


6.6309 


63 


34.6790 


24 


33.7777 


64 


136.0000 


24 


7.1111 


64 


36.0000 


25 


3ftllll 


55 


140.6655 


25 


7.7160 


56 


37.3466 


26 


38.6185 


56 


144.1861 


26 


8.3467 


66 


38.7160 


27 


41.0000 


67 


149.8888 


27 


9.0000 


67 ' 


iO.llll 


28 


43.5556 


68 


154.6666 


28 


9.6790 


68 ' 


11.5308 


29 


46.1861 


59 


169.5186 


29 : 


10.3827 


69 ^ 


12.9753 


30 


48.8888 


60 


1644444 


30 : 


11.1111 60 l( 


14.4444 



1 





( Ixzxix. ) 


BASE 28 SLOPE 2J to 1. 


1 


Add. 


1 


Axld. 


if. of 
Bts. 


Deduct. 


3^ 

^33 


1 

Deduct. , 


K 




K 




1 




Q 


• 


1 


.5647 


31 


60.5647 


.0164 


31 


14.8302 ; 


2 


1.2222 


32 


64.0000 


2 


.0617 


32 


15.8015 


3 


1.9722 


33 


67.K77 


3 


.1389 


33 


1^8055 : 


4 


2.8148 


34 


71.1481 


4 


.2468 


34 


17.8396 


5 


3.7500 


35 


74.8611 


6 


.3867 


36 


18.9043 


6 


4.7777 


36 


78.6666 


6 


.6665 


36 


20.0000 


7 


5.8980 


37 


82.5646 


7 


.7661 


37 


21.1265 


8 


7.1111 


38 


86.6665 


8 


.9876 


38 


22.2839 ■ 


9 


8.4166 


39 


90.6389 


9 


1.2500 


39 


23.4722 


10 


9.8148 


40 


948148 


10 


1.6432 


40 


24.6913 


11 


11.3055 


41 


99.0833 


11 


1.8673 


41 


26.9413 


12 


12.8888 


42 


103.4444 


12 


2.22122 


42 


27.2222 


13 


14.5648 


43 


10?.8981 


13 


2.6080 


43 


28.5360 


14 


16.3333 


44 


112.4444 


14 


3.0247 


44 


29.8765 > 


16 


18.1944 


45 


117.0833 


16 


3.4722 


45 


31.2500 


IB 


20.1481 


46 


121.8148 


16 


3.9506 


46 


32.6543 


17 


22.1944 


47 


126.6389 


17 


4.4599 


47 


340895 


18 


24.3333 


48 


131.6555 


18 


5.0000 


48 


35.5555 


19 


26.5648 


49 


135.6646 


19 


6.6710 


^ 


37.0624 


20 


28.8888 


50 


141.6666 


20 


6.1728 


50 


38.5802 : 


21 


dL3055 


61 


146.8611 


21 


&8055 


61 


40.1388 ■ 


22 


33.8148 


52 


152.1481 


22 


7.4691 


62 


41.7283 ' 


23 


36.4166 


53 


157.6277 


23 


8.1636 


63 


43.3487 


24 


39.1111 


54 


i6aoooo 


24 


8.8889 


54 


46.0000 


25 


41.8980 


65 


168.5647 


25 


9.6455 


66 


4a6820 


26 


44.7777 


66 


173.2222 


26 


10.4321 


56 


48.3950 


27 


47.7500 


57 


179.9722 


27 


11.2500 


67 


6ai388 i 


28 


50.8148 


58 


185.8148 


28 


12.0687 


58 


61.9136 


29 


53.9722 


69 


191.7500 


29 


12.9782 


59 


53.7191 ' 


30 


67-2222 


60 


197.7777 


30 


13.8889 


60 


66.5565 1 



(xc.) 


BASE 28— SLOPE 3 to 1. 


1 

1 


Add. 


4^ 

1 


Add. 


3^ 


Dedact. 




Deduct. 


.5740 


31 


69.4630 


1 


.0185 


31 


17.7963 


2 


1.2592 


32 


73.4815 


2 


.0740 


32 


18.9630 


3 


2.0555 


33 


77.6111 


3 


.1667 


33 


20.1666 


4 


2.9630 


34 


81.8518 


4 


.2963 


34 


21.4074 


5 


3.9815 


35 


86.2037 


6 


.4628 


36 


22.6851 


6 


6.1111 


36 


90.6666 


6 


.6887 


36 


240000 


7 


6.3518 


37 


95.2404 


7 


.9074 


37 


25.3518 


8 


7.7037 


38 


99.9269 


8 


1.1852 


38 


26.7407 


9 


9.1666 


39 


1047222 


9 


1.5000 


39 


28.1666 


10 


10.7407 


40 


109.6296 


10 


1.8518 


40 


29.8296 


11 


12.4260 


41 


114.6481 


11 


2.2407 


41 


31.1296 


12 


14.222?, 


42 


119.7777 


12 


2.6667 


42 


32.8866 


13 


1&1296 


43 


126.0186 


13 


3.1296 


43 


342407 


14 


18.1481 


44 


130.3703 


14 


3.6296 


44 


35.8518 


15 


20.2777 


45 


135.8333 


16 


41667 


46 


37.6000 


16 


22.5185 


46 


141.4074 


16 


47407 


46 


39.1861 


17 


24.8708 


47 


147.0926 


17 


5.3518 


47 


40.9078 


18 


27.3333 


48 


152.8888 


18 


6.0000 


48 


42.6668 


19 


29.9074 


49 


167.7962 


19 


6.6862 


49 


444^29 


20 


32.5926 


50 


164.8148 


20 


7.4074 


60 


46.2963 


21 


35.3888 


61 


170.9444 


21 


8.1687 


51 


49.1666 


22 


38.2962 


52 


177.1861 


22 


8.9629 


52 


60.0740 


23 


41.3148 


53 


183.6370 


23 


9.7962 


63 


52.0185 


24 


44.4444 


54 


190.0000 


24 


10.6667 


64 


540000 


25 


4X6850 


66 


196.5470 


25 


11.6741 


66 


66.0184 


26 


61.0370 


56 


202.2592 


26 


12.5184 


66 


58.0740 


27 


54.5000 


67 


210.0556 


27 


13.5000 


67 


60.1666 


28 


58.0740 


68 


216.9630 


28 


146186 


68 


62.2962 


29 


61.7592 


69 


222.9814 


29 


16.6741 


69 


644639 


30 


65.5655 


60 


231.1111 


30 


16.6667 


60 


66.6666 





(• xci. ) 




BASE 29-SLOPE i to 1. 


•8 


Add. 


•8 


Add. 




Dednct. 


3^ 


Deduct. 


a 




K 




a 




Q 




1 


.5416 


31 


21.0972 


1 


.0015 


31 


1.4830 


2 


1.0926 


32 


21.9259 


2 


.00^ 


32 


1.5802 


8 


1.6527 


33 


22.7639 


3 


.0139 


33 


1.6805 


4 


2.2222 


34 


23.6111 


4 


.0246 


34 


1.7839 


5 


2.8009 


85 


24.4676 


5 


.0385 


35 


1.8904 


6 


3.3888 


36 


26.3333 


6 


.0555 


36 


2.0000 


7 


3.9861 


37 


26.2083 


7 


.0766 


37 


2.1126 


8 


4.5926 


38 


.27.0926 


8 


.0988 


38 


2.2284 


9 


5.2063 


39 


.27.9861 


9 


.1260 


39 


2.3472 


10 


5.8333 


40 


28.8888 


10 


.1543 


40 


2.4691 


11 


&4676 


41 


29.8009 


11 


.1867 


41 


2.6941 


12 


7.1111 


^ 


30.7222 


12 


.2222 


42 


2.7222 


13 


7.7639 


43 


31.6527 


13 


.2608 


43 


2.8545 


14 


8.4259 


44 


32.6926 


14 


.3026 


44 


2.9876 


15 


9.0972 


45 


33.5416 


15 


.3472 


45 


ai250 


16 


9.7777 


46 


345000 


16 


.3961 


46 


3.2654 


17 


10.4676 


47 


35.4676 


17 


.4460 


47 


3.4089 


18 


11.1666 


48 


36.4444 


18 


.5000 


48 


3.5565 


10 


11.8760 


49 


37.4305 


19 


.5671 


49 


3.7052 


20 


12.5926 


50 


38.4259 


20 


.6173 


60 


3.8580 


21 


13.3194 


51 


39.4305 


21 


.6806 


51 


40139 


22 


14.0555 


52 


40.4444 


22 


.7460 


62 


41728 


23 


148009 


53 


41.4676 


23 


.8163 


53 


43349 


24 


15.5555 


54 


42JX)00 


24 


.8889 


54 


46000 


25 


16.3194 


55 


43.5416 


26 


.9647 


55 


46682 


26 


17.0926 


56 


44.6926 


26 


1.0432 


56 


48395 


27 


17.8760 


57 


45.6527 


27 


1.1260 


67 


5.0139 


28 


18.6666 


58 


46.7222 


28 


1.2099 


68 


5.1913 


29 


19.4676 


69 


47.8009 


29 


1.2978 


59 


5.3719 


80 


aO.2777 


60 


48.8888 


30 


1.3889 


60 


6.6555 



1 


( xdL ) 




BASE 29— SLOPE 

r 


; i to 1. 


height 


Add. 


1 


▲dd. 




Deduct. 




Deduct. 


1 




K 




1 




o 
31 




.54^2 


31 


26.6463 


.0031 


2.9660 


2 


1.1111 


32 


26.6666 


2 


.0123 


82 


3.3611 


8 


1.6944 


33 


27.8055 


8 


.0278 


33 


a6111 


4 


2.2962 


34 


28.9629 


4 


.0^4 


84 


3.6679 


5 


2.9166 


35 


30.1388 


5 


.0772 


86 


3.7808 


6 


3.5555 


36 


31.3333 


6 


.1111 


86 


40000 


7 


42129 


37 


32.6463 


7 


.1612 


37 


42258 


8 


48888 


88 


33.7777 


8 


.1975 


38 


4.4668 


9 


5.5833 


39 


36.0277 


9 


.2600 


39 


46944 


10 


6.2962 


40 


36.2962 


10 


.3086 


40 


49383 


11 


7.0278 


41 


37.5833 


11 


.8784 


41 


6.1883 


12 


7.7777 


42 


38.8888 


12 


.4444 


42 


5.4444 


13 


8.5463 


43 


40.2129 


18 


.5216 


48 


6.7076 


14 


9.3333 


44 


41.6566 


14 


.6049 


44 


5.9753 


16 


10.1388 


45 


42.9166 


16 


.6944 


45 


&2500 


16 


10.9629 


46 


4429^ 


16 


.7901 


46 


&5308 


17 


11.8055 


47 


45.6944 


17 


.8920 


47 


6.8179 


18 


12.6666 


48 


47.1111 


18 


1.0000 


48 


7.1111 


19 


18.5463 


49 


^.6462 


19 


1.1142 


49 


7.4105 


20 


144444 


60 


50.0000 


20 


1.2346 


50 


7.7160 


21 


15.8611 


61 


51.4722 


21 


1.3611 


51 


8.02T7 


22 


16.2962 


62 


62.9629 


22 


1.4938 


52 


8.3466 


23 


17.2600 


53 


64.4722 


23 


1.6327 


58 


8.6607 


24 


18.2222 


64 


6aoooo 


24 


1.7778 


54 


9.0000 


25 


19.2129 


56 


57.546@ 


26 


1.9295 


65 


9.3364 


26 


20.2222 


66 


69.1111 


26 


2.0864 


56 


9.6790 


27 


21.2500 


57 


60.6944 


27 


2.2500 


57 


10.0277 


28 


22.2962 


58 


62.2962 


28 


2.4197 


58 


10.3827 


: 29 


23.3611 


59 


63.9166 


29 


2.5966 


59 


10.7488 


! 30 

1 


24.4444 


60 


66.5565 


30 


2,7778 


60 11.1111 





( xciii. ) 




BASE 29 SLOPE i to 1. 


•s 

X 

1 


Add. 


1 


Add. 


Dif.of 
Hts. 


Deduct. 


Dif.of 
Hta. 


Deduct. 


.5508 


31 


29.9964 


1 


.0046 


31 


4.4490 


2 


1.1296 


32 


31.4074 


2 


.0186 


32 


4.7407 


3 


1.7361 


33 


32.8472 


3 


.0416 


33 


5.0416 


4 


2.3703 


34 


343147 


4 


.0740 


34 


5.3518 


5 


3.0323 


35 


35.8101 


6 


.1167 


35 


5.6712 


6 


3.7222 


36 


37.3333 


6 


.1667 


36 


6.0000 


7 


4.4397 


37 


38.8842 


7 


.2268 


37 


6.3379 


8 


6.1851 


38 


40.4629 


8 


.2963 


38 


6.6851 


9 


6.9583 


39 


42.0696 


9 


.3750 


39 


7.0416 


10 


6.7592 


40 


43.7037 


10 


' .4630 


40 


7.4074 


11 


7.5880 


41 


46.3657 


11 


.6602 


41 


7.7824 


12 


8.4444 


42 


47.0555 


12 


.6667 


42 


8.1666 


13 


9.3287 


43 


48.7731 


13 


.7824 


43 


8.5612 


14 


10.2407 


44 


50.5185 


14 


.9074 


44 


8.9629 


15 


11.1806 


45 


52.2916 


15 


1.0417 


45 


9.3750 


16 


12.1481 


46 


54.0926 


16 


1.1852 


46 


9.79^ 


17 


13.1434 


47 


56.9212 


17 


1.3379 


47 


10.2268 


18 


14.1666 


48 


67.7777 


18 


1.5000 


48 


10.6660 


19 


15.2176 


^ 


59.6620 


19 


1.6713 


49 


11.1167 


20 


16.2962 


50 


61.6740 


20 


1.8618 


50 


11.5740 


21 


17.4027 


61 


63.5140 


21 


2.0417 


51 


12.0416 


22 


18.5370 


52 


66.4814 


22 


2.2407 


62 


12.5185 


23 


19.6991 


53 


67.4768 


23 


2.4491 


53 


13.0046 


24 


20.8888 


64 


69.5000 


24 


2.6667 


64 


13.5000 


25 


22.1064 


65 


71.6608 


26 


2.8935 


56 


14.0046 


26 


23.3518 


56 


73.6296 


26 


3.1296 


56 


14.5185 


27 24.6250 


67 


76.7361 


27 


3.3750 


57 


15.0416 


28 


25.9258 


58 


77.8703 


28 


3.6296 


58 


15.67^ 


29 27.2546 


69 


80.0323 


29 


3.8986 


5e 


16.1157 


30 28.6111 


60 


82.2222 


30 


4.1667 


60 


16.6666 





( xciv. ) 




BASE 29— SLOPE 1 to 1. 


1 


Add. 


} 


Add. 




Deduct. 


ftsSJ 


Deduct. 


to 




B 




a 




Q 




1 


.5555 


31 


34.4444 


1 


.0062 


31 


6.9321 


2 


1.1481 


32 


36.1481 


2 


.0247 


32 


&3210 


3 


1.7777 


33 


37.8888 


3 


.0556 


33 


6.7222 


4 


2.4444 


34 


39.6666 


4 


.0988 


34 


7.1358 


5 


ai481 


35 


41.4814 


5 


.1643 


35 


7.5617 


6 


a8888 


36 


43.3333 


6 


.2222 


36 


8.0000 


7 


46666 


37 


45.2222 


7 


.3025 


37 


8.4506 


8 


5.4814 


38 


47.1481 


8 


.3951 


38 


8.9135 


9 


6.3333 


39 


49.1111 


9 


.5000 


39 


9.3888 


10 


7.2222 


40 


61.1111 


10 


.6178 


40 


9.8765 


11 


ai481 


41 


53.1481 


11 


.7469 


41 


10.3765 


12 


9.1111 


42 


55.2222 


12 


.8889 


^ 


10.8888 


13 


10.1111 


43 


67.3333 


13 


1.0432 


43 


11.4135 


14 


11.1481 


44 


69.4814 


14 


1.2099 


44 


11.9506 


15 


12.2222 


45 


61.6666 


16 


1.3889 


46 


12.6000 


16 


13.3333 


46 


64.8888 


16 


1.5802 


46 


13.0617 


17 


14.4814 


47 


66.1481 


17 


1.7839 


47 


ia6358 


18 


15.6666 


48 


68.4444 


18 


2.0000 


48 


14.2222 


19 


16.8888 


49 


70.7777 


19 


2.2284 


49 


14.8209 


20 


18.1481 


50 


73.1481 


20 


2.4691 


60 


15.4321 


21 


19.4444 


51 


76.5555 


21 


2.7222 


51 


16.0565 


22 


20.7777 


62 


78.0000 


22 


2.9876 


52 


16.6913 


23 


22.1481 


53 


80.4814 


23 


a2664 


63 


17.3395 


24 


23.5555 


54 


82.0000 


24 


3.5566 


64 


18.0000 


26 


25.0000 


55 


85.5665 


26 


a8630 


55 


ia6728 


26 


26.4815 


56 


88.1481 


26 


4.1728 


56 


19.3580 


27 


28.0000 


67 


90.7777 


27 


4.5000 


57 


20.0555 


28 


29.5555 


68 


93.4444 


28 


4.8395 


58 


20.7654 


29 


31.1481 


59 


96.1481 


29 


5.1913 


69 


21.4876 


30 


32.7777 


60 


98.8888 


30 


5.6655 


60 







( «T. ) 


BASE 29— SLOPE li to 1. 


1 

•8 


Add. 


* 


Add. 


if. of 


Deduct. 


if. of 
Hti. 


Deduct. 


X 
1 




31 




Q 




Q 




.5601 


38.8935 


1 


.0077 


31 


7.4161 


2 


1.1666 


32 


40.8888 


2 


.0309 


32 


7.9012 


3 


1.8194 


33 


^.9305 


3 


.0694 


33 


8.4027 


4 


2.6185 


34 


45.0185 


4 


.1234 


34 


8.9197 


5 


3.2638 


35 


47.1527 


5 


.1928 


35 


9.4621 


6 


4.0555 


36 


49.3333 


6 


.2778 


36 


10.0000 


7 


4.8934 


37 


51.6601 


7 


.3781 


37 


10.5632 


8 


5.7777 


38 


63.8333 


8 


.4938 


38 


11.1419 


9 


6.7083 


39 


66.1527 


9 


.6250 


39 


11.7361 


10 


7.6851 


40 


58.5185 


10 


.7716 


40 


12.3456 


11 


8.7083 


41 


60.9305 


11 


.9336 


41 


12.9706 


12 


9.7777 


42 


63.3888 


12 


1.1111 


^ 


13.6111 


13 


10.8934 


43 


66.8936 


13 


1.3040 


43 


14268G 


14 


12.0555 


44 


68.4444 


14 


1.5123 


44 


149382 


15 


13.2638 


45 


71.0416 


15 


1.7361 


45 


15.6250 


16 


14.5185 


46 


74.6851 


16 


1.9753 


46 


16.3271 


17 


15.8194 


47 


76.3750 


17 


2.2299 


47 


17.0447 


18 


17.1666 


48 


79.1111 


18 


2.5000 


48 


17.7777 


19 


18.5601 


49 


81.8935 


19 


2.7855 


49 


18.5262 


20 


20.0000 


50 


84.7222 


20 


3.0864 


50 


19.2901 


21 


21.4861 


61 


87.5972 


21 


3.4028 


61 


20.0694 


22 


23.0185 


52 


90.5186 


22 


3.7346 


52 


20.8641 


23 


245972 


63 


93.4860 


23 


4.0818 


53 


21.6743 


24 


26.2222 


54 


95.5000 


24 


4.4444 


54 


22.6000 


25 


27.8935 


55 


99.5601 


25 


4.8228 


56 


23.3410 


26 


29.6111 


56 


102.6666 


26 


6.2160 


66 


24.1975 


27 


31.3750 


57 


106.8194 


27 


5.6260 


57 


25.0694 


28 


33.1851 


68 


109.0186 


28 


6.0494 


58 


25.9667 


29 


35.0416 


59 


112.2638 


29 


6.4891 


59 


26.8595 


80 


36.9444 


60 


116.5555 


30 


6.9444 


60 


27.7777 







( xcvi. ) 




BASE 29— SLOPE IJ to 1. 


1 


Add. 


1 


Add. 


^9 


Dednct. 


SIS 


Deduct. 


V 




s 




Q 




Q 




1 


.6647 


31 


43.3426 


1 


.0092 


31 


8.8981 


2 


1.1851 


32 


46.6296 


2 


.0370 


32 


9.4816 


3 


1.8611 


33 


47.9722 


3 


.0833 


33 


10.0833 


4 


2.5926 


34 


50.3703 


4 


.1480 


34 


10.7037 


5 


3.3796 


35 


62.8240 


6 


.2313 


35 


11.3426 


6 


42222 


36 


66.3333 


6 


.3333 


36 


12.0000 


7 


6.1202 


37 


67.8980 


7 


.4537 


37 


12.6769 


8 


6.0740 


38 


60.5186 


8 


.5926 


38 


13.3703 


9 


7.0833 


39 


63.1944 


9 


.7500 


39 


140833 


10 


8.1481 


40 


66.9269 


10 


.9259 


40 


148148 


11 


9.2685 


41 


08.7129 


11 


1.1203 


41 


16.6648 


12 


10.4444 


42 


71.6655 


12 


1.3333 


42 


ia3333 


13 


11.6758 


43 


74.4537 


13 


1.5648 


43 


17.1202 


14 


12.9629 


44 


77.4074 


14 


1.8148 


44 


17.9269 


15 


14.3065 


46 


80.4166 


15 


2.0833 


46 


18.7500 


16 


15.7037 


46 


84.4814 


16 


2.3704 


46 


19.6925 


17 


17.1573 


47 


86.6018 


17 


2.6750 


47 


20.4637 


18 


18.6666 


48 


89.7777 


18 


3.0000 


48 


21.3333 


19 


20.2314 


49 


93.0092 


19 


3.3426 


49 


22.2314 


20 


21.8618 


60 


96.29^ 


20 


3.7037 


50 


23.1481 


21 


23.5277 


61 


99.6388 


21 


4.0833 


51 


24.0833 


22 


26.2592 


62 


103.0370 


22 


44815 


52 


25.0370 


23 


27.0462 


63 


106.4906 


23 


48981 


53 


26.0092 


24 


28.8888 


64 


109.0000 


24 


5.3333 


64 


27.0000 


26 


30.7870 


55 


113.6647 


25 


6.7869 


56 


28.0092 


26 


32.7407 


66 


117.1852 


26 


6.2592 


66 


29.0370 


27 


34.7500 


67 


120.8611 


27 


6.7600 


57 


30.0833 


28 


36.8147 


68 


124.6926 


28 


7.2592 


58 


31.1481 


29 


38.9361 


69 


128.3796 


29 


7.7869 


59 


32.2314 


30 


41.1111 


60 


132.2222 


30 8.33331 


60 


3a3333 





( xcvii. ) 




BASE 29— SLOPE If to 1. 


Height 


Add. 


■s 

» 


Add. 


1 


Deduct. 


Dif. of 
Hts. 


Deduct. 


1 


.6692 


31 


47.7917 


.0108 


31 


10.3811 


2 


1.2037 


32 


50.3708 


2 


.0482 


82 


11.0617 


3 


1.9028 


33 


63.0140 


3 


.0972 


38 


11.7638 


4 


2.6666 


84 


65.7222 


4 


.1728 


84 


12.4876 


5 


3.4952 


36 


58.4968 


6 


.2700 


85 


18.2830 


6 


4.3888 


36 


61.3383 


6 


.8889 


86 


14.0000 


7 


5.3470 


37 


64.2360 


7 


.6293 


37 


14.7885 


8 


6.37^ 


88 


67.2087 


8 


.6913 


38 


15.5987 


9 


7.4583 


39 


70.2360 


9 


.8760 


89 


16.4305 


10 


8.6111 


40 


78.8388 


10 


1.0802 


40 


17.2839 


11 


9.8286 


41 


76.4958 


11 


1.3071 


41 


18.1689 


12 


11.1111 


42 


79.7222 


12 


1.6666 


42 


19.0555 


13 


12.4582 


43 


88.0140 


18 


1.8256 


43 


19.9747 


14 


13.8708 


44 


86.8703 


14 


2.1173 


44 


20.9185 


15 


16.8470 


46 


89.7917 


15 


2.4305 


45 


21.8750 


16 


1&8888 


46 


94.2777 


16 


2.7654 


46 


22.8580 


17 


18.4962 


47 


96.8286 


17 


3.1219 


47 


28.8626 


18 


20.1666 


48 


100.4444 


18 


8.5000 


48 


24.8888 


19 


21.9027 


49 


104.1250 


19 


8.8997 


49 


26.9867 


20 


23.7037 


60 


107.8708 


20 


4.8210 


50 


27.0061 


21 


25.6694 


51 


111.6805 


21 


4.7639 


51 


28.0972 


22 


27.6000 


52 


115.5655 


22 


5.2284 


52 


29.2098 


23 


29.4952 


53 


119.4962 


23 


5.7146 


63 


30.8441 


24 


31.5566 


64 


122.6000 


24 


6.2222 


54 


31.6000 


25 


33.6806 


65 


127.5694 


25 


6.7516 


65 


32.6774 


26 


86.8703 


66 


131.7037 


26 


7.8024 


66 


83.8765 


27 


38.1260 


57 


ia5.9027 


27 


7.8750 


67 


36.0972 


28 


40.4444 


58 


140.1666 


28 


8.4691 


58 


86.8394 


29 


42.8286 


69 


144.4962 


29 


9.0848 


69 


37.6033 


80 


46.2777 


60 


148.8888 


30 


9.7222 


60 


38.8888 

1 
I 





( xcviii. ) 




BASE 29-SLOPE 2 to 1. 




t 


Add. 


•s 


Add. 


Si as 


Deduct. 


5S3 


Deduct. 




» 




cs 




1 




Q 






1 


.6740 


31 


62.2407 


.0123 


31 


11.8642 




2 


1.2222 


32 


55.1111 


2 


.0494 


32 


12.6419 




3 


1.9444 


33 


68.0566 


3 


.1111 


33 


13.4444 




4 


2.7407 


34 


61.0740 


4 


.1975 


34 


14.2716 




5 


3.6111 


36 


64.1666 


6 


.3086 


36 


16.1234 




6 


4.6555 


36 


67.3333 


6 


.4444 


36 


16.0000 




7 


5.5740 


37 


70.6740 


7 


.6049 


87 


16.9012 




8 


6.6666 


38 


73.8888 


8 


.7901 


38 


17.8271 




9 


7.8333 


39 


77.2777 


9 


1.0000 


89 


18.7777 




10 


9.0740 


40 


80.7407 


10 


1.2346 


40 


19.7530 




11 


10.3888 


41 


842777 


11 


1.4938 


41 


20.7630 




12 


11.7777 


42 


87.8888 


12 


1.7778 


42 


21.7777 




13 


13.1406 


43 


91.5740 


13 


2.0864 


43 


22.8271 




14 


14.7777 


44 


95.3333 


14 


2.4197 


44 


23.9012 


• 


15 


16.3888 


45 


99.1666 


15 


2.7778 


46 


25.0000 




16 


18.0740 


46 


104.0740 


16 


3.1606 


46 


26.1234 




17 


19.8333 


47 


107.0555 


17 


3.6679 


47 


27.2716 




18 


21.6666 


48 


iii.iin 


18 


4.0000 


48 


28.4444 




19 


23.5740 


49 


115.2407 


19 


4.4568 


49 


29.6^20 




20 


25.5555 


50 


119.4444 


20 


4.9382 


60 


80.8642 




21 


27.6111 


61 


123.7222 


21 


5.4444 


51 


32.1111 




22 


29.7407 


52 


128.0740 


22 


5.9753 


62 


33.3827 




23 


31.9444 


53 


182.6000 


23 


6.6809 


53 


34.6790 




24 


34.2222 


54 


136.0000 


24 


7.1111 


54 


36.0000 




25 


36.6740 


65 


141.6740 


25 


7.7160 


66 


37.8466 




26 


39.0000 


56 


146.2222 


26 


8.3457 


56 


88.7160 




27 


41.5000 


67 


160.9444 


27 


9.0000 


57 


40.1111 




28 


44.0740 


58 


166.7407 


28 


9.6790 


68 


41.5808 




29 


46.7222 


59 


160.6111 


29 


10.3827 


69 


42.9763 




30 


49.4444 


60 


165.5556 


80 


11.1111 


60 


44.4444 









( xctx. ) 








BASE 29 SLOPE 


2| to 1. 


m 

1 


Add. 


1 

» 


Add. 




Deduct. 




Deduct. 


.5833 


31 


61.1388 


1 


.0164 


31 


14.8302 


2 


1.2592 


32 


64.6926 


2 


.0617 


32 


16.8013 


8 


2.0277 


33 


68.1389 


3 


.1389 


S3 


ia8055 


4 


2.8888 


34 


71.7777 


4 


.2468 


34 


17.8395 


5 


3.8426 


35 


76.5092 


5 


.3867 


36 


18.9043 


6 


4.8888 


36 


79.3333 


6 


.5665 


36 


20.0000 


7 


6.0277 


37 


83.2600 


7 


.7561 


37 


21.1265 


8 


7.2592 


38 


87.2592 


8 


.9876 


38 


22.2839 


9 


8.5833 


39 


91.3611 


9 


1.2500 


39 


23.4722 


10 


10.0000 


40 


96.6665 


10 


1.5432 


40 


246913 


H 


11.5092 


41 


99.8426 


11 


1.8673 


41 


25.9413 


12 


13.1111 


42 


104.2222 


12 


2.2222 


42 


27.2222 


13 


14.8054 


43 


108.6944 


13 


2.6080 


43 


28.5360 


14 


16.5926 


44 


113.2592 


14 


3.0247 


44 


29.8765 


16 


18.4722 


45 


117.9166 


16 


3.4722 


46 


31.2600 


16 


20.4444 


46 


123.6666 


16 


3.9506 


46 


32.6543 


17 


22.5092 


47 


127.5092 


17 


4.4599 


47 


340896 


18 


24.6666 


48 


132.4444 


18 


5.0000 


48 


36.6655 


19 


26.9166 


49 


137.4722 


19 


6.5710 


49 


37.0524 


20 


29.2592 


50 


142.6926 


20 


6.1728 


50 


3a6802 


21 


31.6944 


51 


147.8066 


21 


6.8056 


51 


40.1388 


22 


34.2222 


52 


153.1111 


22 


7.4691 


52 


41.7283 


23 


36.8426 


53 


158.5092 


23 


8.1636 


53 


43.3487 


24 


39.5555 


54 


163.0000 


24 


8.8889 


54 


45.0000 


25 


42.3610 


55 


169.5833 


25 


9.6466 


55 


46.6820 


26 


45.2592 


56 


175.2592 


26 


10.4321 


56 


48.3950 


27 


48.2500 


57 


181.0277 


27 


11.2500 


67 


60.1388 


28 


51.3333 


68 


186.8888 


28 


12.0687 


68 


51.9135 


29 


54.5092 


59 


192.8425 


29 


12.9782 


69 


53.7191 


30 


57.7777 


60 


198.8888 


30 


13.8889 


60 


65.6656 



(c) 




BASE 29— SLOPE 3 to 1. 


•4* 

1 
1 


Add. 


1 


Add. 


3^ 

1 


Deduct. 


I>if.of 

Ht8. 


Deduct. 


.&926 


31 


70.0370 


.0185 


31 


17.796a 


2 


1.2962 


32 


740740 


2 


.0740 


32 


18.9630 


a 


2.1111 


33 


78.2222 


3 


.1667 


33 


20.1666 


4 


3.0370 


34 


82.4814 


4 


.2963 


34 


21.4074 


6 


40740 


35 


86.8618 


5 


.4628 


35 


22.6851 


6 


5.2222 


36 


91.3333 


6 


.6667 


36 


240000 


7 


6.4813 


37 


96.9268 


7 


.9074 


37 


25.3518 


8 


7.8618 


38 


100.6296 


8 


1.1862 


38 


26.7407 


9 


9.3333 


39 


106.4444 


9 


1.6000 


39 


28.1666 


10 


10.9259 


40 


110.3703 


10 


1.8618 


40 


29.6296 


11 


12.6296 


41 


116.4074 


11 


2.2407 


41 


31.1296 


12 


14.4444 


42 


120.5555 


12 


2.6667 


42 


32.6666 


13 


16.3703 


43 


126.8148 


13 


3.1296 


43 


34.2407 


14 


18.4074 


44 


131.1861 


14 


3.6296 


44 


35.8518 


15 


20.5555 


45 


136.6666 


15 


41667 


45 


37.5000 


16 


22.8148 


46 


143.2692 


16 


47407 


46 


39.1861 


17 


25.1851 


47 


147.9628 


17 


6.3618 


47 


40.9074 


18 


27.6666 


48 


163.7777 


18 


6.0000 


48 


42.6666 


19 


30.2592 


49 


169.7037 


19 


6.6862 


49 


44.4629 


20 


32.9629 


50 


165.7407 


20 


7.4074 


60 


46.2963 


21 


36.7777 


51 


171.8888 


21 


8.1667 


51 


49.1666 


22 


38.7037 


62 


178.1481 


22 


8.9629 


62 


50.0740 


23 


41.7407 


63 


1845186 


23 


9.7962 


53 


52.0185 


24 


44.8888 


54 


190.0000 


24 


10.6667 


54 


54.0000 


25 


48.1480 


65 


197.5926 


25 


11.6741 


65 


56.0184 


26 


61.5184 


56 


2042962 


26 


12.5184 


56 


58.0740 


27 


55.0000 


67 


211,1111 


27 


13,5000 


57 


60.1666 


28 


68.5926 


58 


218.0370 


28 


145186 


68 


62.2962 


29 


62.2962 


59 


226.0740 


29 


15.5741 


69 


64.4629 


30 


66.1111 


60 


232.2222 


30 


16.6667 


60 


66.6666 





(oi.) 


BASE 30— SLOPE i to 1. 


V 


Add. 


1 


Add. 




Deduct. 




Deduct. 


» 




X 




1 




31 




1 


.6602 


31 


21.6713 


.0015 


1.4830 


2 


1.1296 


32 


22.5186 


2 


.0062 


32 


1.5802 


3 


1.7083 


33 


23.3750 


3 


.0139 


33 


1.6805 


4 


2.2962 


34 


24.2407 


4 


.0246 


34 


1.7839 


5 


2.8935 


35 


25.1157 


5 


.0385 


35 


1.8904 


6 


3.6000 


36 


26.0000 


6 


.0565 


36 


2.0000 


7 


4.1157 


37 


26.8935 


7 


.0766 


37 


2.1126 


8 


4.7407 


38 


27.7963 


8 


.0988 


38 


2.2284 


9 


5.3750 


39 


28.7083 


9 


.1250 


39 


2.3472 


10 


6.0185 


40 


29.6296 


10 


.1543 


40 


2.4691 


11 


6.6713 


41 


30.5602 


11 


.1867 


41 


2.5941 


12 


7.3333 


42 


31.6000 


12 


.2222 


42 


2.7222 


13 


8.0046 


43 


32.4490 


13 


.2608 


43 


2.8545 


14 


8.6851 


44 


33.4073 


14 


.30-25 


44 


2.9876 


15 


9.3760 


45 


34.3750 


15 


.3472 


45 


3.1250 


16 


10.0740 


46 


35.3518 


16 


.3961 


46 


a2654 


17 


10.7824 


47 


36.3379 


17 


.4460 


47 


3.4089 


18 


11.5000 


48 


37.3333 


18 


.6000 


48 


3.5555 


19 


12.^268 


49 


38.3379 


19 


.5.571 


49 


3.7052 


20 


12.9629 


60 


39.3518 


20 


.6173 


60 


3.8580 


21 


13.7083 


61 


40.3750 


21 


.6805 


51 


4.0139 


22 


14.4629 


62 


41.4073 


22 


.7469 


52 


4.1728 


23 


15.2268 


63 


42.4490 


23 


.8163 


53 


4.3349 


24 


16.0000 


54 


43.5000 


24 


.8889 


54 


4.5000 


25 


16.7824 


65 


44.5602 


25 


.9647 


55 


4.6682 


26 


17.5740 


56 


46.6296 


26 


1.0432 


66 


4.8395 


27 


18.3760 


67 


46.7083 


27 


1.1250 


67 


5.0139 


28 


19.1851 


58 


47.7963 


28 


1.2099 


58 


5.1913 


29 


20.0046 


59 


48.8935 


29 


1.2978 


59 


5.3719 


30 


20.8333 


60 


60.0000 


30 


1.3889 


60 


5.5555 





( cii. ) 




BASE 30— SLOPE i to 1. 




•a 

•8 


Add. 


1 


Add. 


Saru 


Deduct. 




Deduct. 




1 




31 




1 




a^ 






.6648 


26.1203 


.0031 


31 


2.9660 




2 


1.1481 


32 


27.2592 


2 


.0123 


32 


3.1605 




3 


1.7600 


33 


28.4166 


3 


.0278 


33 


33611 




4 


2.3704 


34 


29.6926 


4 


.0494 


34 


3.6679 




3 


3.0092 


36 


30.7870 


5 


.0772 


36 


3.7808 




6 


3.6666 


36 


32.0000 


6 


.1111 


36 


4.0000 




7 


4.3426 


37 


33.2316 


7 


.1612 


37 


4.2263 




8 


6.0370 


38 


34.4816 


8 


.1975 


38 


4.4668 




9 


6.7600 


39 


36.7600 


9 


.2600 


39 


4.6944 




10 


6.4816 


40 


37.0370 


10 


.3086 


40 


4.9383 




11 


7.23J6 


41 


38.3425 


11 


.3734 


41 


6.1883 




12 


8.0000 


42 


39.6666 


12 


.4444 


42 


5.4444 




13 


8.7870 


43 


41.0092 


13 


.6216 


43 


6.7067 




14 


9.6926 


44 


42.3704 


14 


.6049 


44 


5.9753 




16 


10.4166 


46 


43.7500 


15 


.6944 


46 


6.2600 




16 


11.2692 


46 


46.1481 


16 


.7901 


46 


6.6308 




17 


12.1203 


47 


46.6648 


17 


.8920 


47 


6.8179 




18 


13.0000 


48 


48.0000 


18 


1.0000 


48 


7.1111 




19 


13.8981 


49 


49.4637 


19 


1.11^ 


49 


7.4106 




20 


14.8147 


60 


60.9260 


20 


1.2346 


60 


7.7160 




21 


16.7600 


61 


52.4166 


21 


1.3611 


61 


8.0277 


• 


22 


16.7037 


62 


63,9260 
56.4637 


22 


1.4938 


62 


8.3466 




23 


17.6759 


63 


23 


1.6327 


63 


8.6697 




24 


18.6666 


64 


67.0000 


24 


1.7778 


54 


9.0000 




26 


19.6769 


65 


58.5648 


26 


1.9296 


65 


9.3364 




26 


20.7037 


66 


60.1481 


26 


2.0864 


56 


9.6790 




27 


21.7500 


57 


61.7600 


27 


2.26()0 


67 


10.0277 




28 


22.8147 


68 


63.3704 


28 


2.4197 


68 


10.3827 




29 


23.8981 


69 


65.0092 


29 


2.6956 


69 


10.7438 


1 
• 


30i26.0000 


60 


66.6666 


30 


2.7778 


60 


11.1111 





( ciiL ) 
BASE 30— SLOPE | to 1. 



-a, 

•s 



Add. 



1 
2 
3 
4 
6 

6 
7 
8 
9 
10 

11 
12 
13 
14 
15 

16 
17 
18 
19 
20 

21 
22 
23 
24 
26 

26 
27 
28 
29 
30 



•s 



.5694 
1.1666 
1.7917 
2.4444 
3.1250 

3.8333 
4.5692 
5.3333 
6.1250 
6.9444 

7.7917 

8.6666 

9.5694 

10.5000 

1L4583 

12.4444 
13.4583 
14.5000 
15.5694 
16.6666 

17.7917 
18.9444 
20.1250 
21.3333 
22.5692 

23.8333 
25.1250 
26.4444 
27.7917 
29.1666 



Add. 



31 
32 
33 
34 
35 

36 
37 

38 
39 
40 

41 
42 
43 
44 
45 

46 
47 

48 
49 
50 

61 
52 
63 
54 
56 

66 
67 

68 
69 
60 



30.5694 
32.0000 
33.4683 
34.9444 
36.4583 

38.0000 
39.6694 
41.1666 
42.7917 
44.4444 

46.1250 
47.8333 
49.6692 
51.3333 
63.1250 

64.9444 
66.7917 
58.6666 
60.5694 
62.5000 

64.4683 
66.4444 
68.4583 
70.5000 
72.6694 

74.6666 
76.7917 
78.9444 
81.1250 
83.3333 



o • 



Deduct. 



1 

2 
3 
4 
5 

6 

7 

8 

9 

10 

11 
12 
13 
14 
16 

16 
17 
18 
19 
20 

21 
22 
23 
24 
26 

26 
27 
28 
29 
30 



i 



s 



.0046 
.0185 
.0416 
.0740 
.1167 

.1667 
.2268 
.2963 
.3760 
.4630 

.5602 
.6667 
.7824 
.9074 
1.0417 

1.1852 
1.3379 
1.5000 
1.6713 
1.8518 

2.0417 
2.2407 
2.4491 
2.6667 
2.8935 

3.1296 
3.3750 
3.6296 
3.8936 
4.1667 



31 
32 
33 
34 
35 

36 
37 
38 
39 
40 

41 
42 
43 
44 
46 

46 
47 
48 
49 
50 

51 
62 
63 
54 
55 

56 
51 
68 
59 
60 



Deduct. 

4.4490 
4.7407 
5.0416 
6.3618 
66712 

6.0000 
6.3379 
6.6851 
7.0416 
7.4074 

7.7824 
8.1666 
8.5602 
8.9629 
9.3750 

9.7962 
10.2268 
10.6666 
11.1157 
11.5740 

12.0416 
12.5185 
13.0046 
13.5000 
14.0046 

14.6186 
15.0416 
15.6740 
16.1157 
16.6666 





( «T. ) 




BASE 30— SLOPE 1 to 1. 




•8 


Add. 


to 

■s 


Add. 




Deduct. 




Dedoct. 




S 




w 




1 




Q 






1 


.5740 


31 


35.0186 


.0062 


31 


5.9321 




2 


1.1861 


32 


36.7407 


2 


.0247 


32 


6.321U 




3 


1.8333 


33 


88.5000 


3 


.05.55 


33 


6.7222 


• 


4 


2.6186 


34 


40.2962 


4 


.0988 


34 


7.1358 




5 


3.2407 


36 


42.1296 


5 


.1643 


36 


7.6617 




6 


4.0000 


36 


44.0000 


6 


.2222 


36 


8.0000 




7 


4.7962 


37 


46.9074 


7 


.3025 


37 


8.4606 




8 


6.6296 


38 


47.8518 


8 


.3951 


38 


8.9136 




9 


6.5000 


39 


49.8333 


9 


.6000 


39 


9.3888 




10 


7.4074 


40 


51.8618 


10 


.6173 


40 


9.8766 




11 


8.3618 


41 


53.9074 


11 


.7469 


41 


10.3765 




12 


9.3333 


42 


66.0000 


12 


.8889 


42 


10.8888 




13 


10.3618 


43 


68.1296 


13 


1.0432 


43 


11.4135 




14 


11.4074 


44 


60.2962 


14 


1.2099 


44 


11.9506 




15 


12.6000 


45 


62.6000 


15 


1.3889 


46 


12.6000 




16 


13.^96 


46 


64.7407 


16 


1.6802 


46 


13.0617 




17 


14.7962 


47 


67.0186 


17 


1.7839 


47 


13.6358 




18 


16.0000 


48 


69.3333 


18 


2.0000 


48 


14.2222 




19 


17.2407 


49 


71.6851 


19 


2.2284 


49 


14.8209 




20 


18.5186 


60 


74.0740 


20 


2.4691 


60 


15.4321 




21 


19.8.333 


61 


76.6000 


21 


2.7222 


51 


16.0555 


1 


22 


21.1851 


52 


78.9629 


22 


2.9876 


62 


16.6913 




23 


22.6740 


53 


81.4629 


23 


3.2654 


63 


17.3396 




24 


24.0000 


54 


84.0000 


24 


3.6656 


54 


18.0000 




26 


26.4629 


66 


86.6740 


25 


3.8580 


55 


18.6728 




26 


26.9629 


66 


89.1851 


26 


4.1728 


66 


19.3680 




27 


28.6000 


57 


91.8333 


27 


4.5000 


57 


20.0666 




28 


30.0740 


68 


94..5185 


28 


4.8396 


58 


20.7654 


1 


29 


31.6861 


59 


97.2407 


29 


6.1913 


69 


21.4876 




30 


33.3333 


60 


100.0000 1 30 


5.5665 60 


22.2222 







( cv. ) 


BASE 30— SLOPE IJ to 1. 


S 




S 




0.5I 


"8 J 




X 


AM. 




Add. 


. .3 


Dedoek 


2-« 


Dedoot. 


1 


.6786 


31 


39.4676 


.0077 


31 


7.4151 


2 


1.2037 


32 


41.4815 


2 


.0809 


32 


7.9012 


3 


1.8750 


33 


43.5417 


8 


.0694 


38 


8.4027 


4 


2.5926 


34 


45.6481 


4 


.1234 


34 


8.9197 


5 


3.3564 


36 


47.8009 


6 


.1928 


35 


9.4521 


6 


4.1666 


36 


50.0000 


6 


.2778 


36 


10.0000 


7 


5.0230 


37 


52.2453 


7 


.3781 


37 


10.6632 


6 


5.9259 


38 


64.5370 


8 


.4938 


38 


11.1419 


9 


6.8750 


39 


56.8760 


9 


-6260 


39 


11.7361 


10 


7.8703 


40 


69.2992 


10 


.7716 


40 


12.3466 


11 


8.9120 


41 


61.6898 


11 


.9336 


41 


12.9706 


12 


10.0000 


42 


64.1666 


12 


1.1111 


42 


laoiii 


13 


11.1344 


43 


66.6898 


13 


1.3040 


43 


14.2680 


14 


12.3148 


44 


69.2592 


14 


1.5123 


44 


14.9382 


15 


13.5416 


45 


71.8760 


15 


1.7361 


46 


15.6250 


}6 


14.8148 


46 


74.5370 


16 


1.9753 


46 


16.3271 


17 


16.1344 


47 


77.2463 


17 


2.2299 


47 


17.0447 


IS 


17.5000 


48 


80.0000 


18 


2.6000 


48 


17.7777 


19 


19.9120 


49 


^.8009 


19 


2.7855 


49 


18.5262 


«) 


20.3703 


50 


85.6481 


20 


3.0864 


50 


19.2901 


21 


21.8750 


51 


8a6417 


21 


3.4028 


51 


20.0694 


i» 


23.^59 


52 


91.4816 


22 


3.7346 


52 


20.8641 


23 


26.0230 


53 


94.4676 


23 


4.0818 


53 


21.6743 


24 


26.6666 


54 


97.6000 


24 


4.4444 


54 


22.5000 


25 


28.3564 


55 


100.5786 


25 


4.8228 


55 


S3.3410 


26 


aoxme 


56 


108.7037 


26 


5.2160 


56 


24.1976 


27 


31.8750 


67 


106.8750 


27 


6.^50 


57 


25.0694 


28 


33.7037 


58 


110.0926 


28 


6.0494 


58 


25.9667 


29 


35.5786 


59 


113.3564 


29 


6.4891 


59 


26.8696 


30 


^.5000 


60 


116.6666 


30 


6.9444 


60 
p 


27-7777 

1 



( CT>- ) 

BASE 30— SLOPE 1^ to 1. 



I 

» 



1 
2 
3 

4 
6 

6 
7 
8 
9 
10 

11 
12 
13 
14 
15 

16 
17 
18 
19 
20 

21 
22 
23 
24 
25 

26 
27 
28 
29 
30 



Add. 



.6833 
1.2222 
1.9166 
2.6666 
a4722 

43333 
5.2600 
6.2222 
7.2600 
8.3333 

9.4722 
10.6666 
11.9166 
13.2222 
14.5833 

16.0000 
17.4722 
19.0000 
21.5833 
22.2222 

23.9166 
25.6666 
27.4722 
29.3333 
31.2500 

33.2222 
35.2500 
37.3333 
39.4722 
41.6666 



I 



31 
32 
33 
34 
36 

36 

37 
38 
39 
40 

41 
42 
43 
44 
45 

46 
47 

48 
49 
50 

51 
52 
53 
54 
55 

56 

57 
58 
69 
60 



Add. 



43.9166 
46.2222 
48.5833 
51.0000 
53.4722 

66.0000 
58.6833 
61.22^ 
63.9166 
66.6666 

69.4722 
72.3333 
75.2500 
78.2222 
81.2500 

843333 
87.4722 
90.6666 
93.9166 
97.2222 

100.5833 
1040000 
107.4722 
111.0000 
1145833 

118.2222 
121.9166 
125.6666 
129.4722 
133.3333 



•s 



» 



1 
2 
3 

4 
6 

6 
7 
8 
9 
10 

11 
12 
13 
14 
15 

16 
17 
18 
19 
20 

21 
22 
23 
24 
25 

26 
27 
28 
29 
30 



Deduct. 



.0092 
.0370 
.0833 
.1480 
.2313 

.3333 
.4537 
.6926 
.7500 
.9259 

1.1208 
1.3333 
1.5648 
1.8148 
2.0833 

2.3704 
2.6769 
3.0000 
3.3426 
3.7037 

4.0633 
4.4815 
4.8981 
5.3333 
5.7869 

6.2592 
6.7500 
7.2592 
7.7869 
a3333 



O • 



31 
32 

33 
34 
36 

36 
87 
88 
39 
40 

41 
42 
43 
44 

45 

46 
47 
48 
49 
50 

61 
62 
63 
54 

65 

66 
57 
58 
59 
60 



Deduct. 

8.8981 

9.4816 

10.0833 

10.7037 

11.3426 

12.0000 
12.6759 
13.3708 
14.0833 
14.8148 

16.6648 
16.3333 
17.1202 
17.9269 
18.7600 

19.6926 
20.4637 
21.3333 
22.2314 
23.1481 

24.0833 
25.0370 
26.0092 
27.0000 
28.0092 

29.0370 
30.0833 
31.1481 
32.2314 
33.3333 



( CTii. ). 


BASE 80— SLOPE U to 1. 


1 


KM. 


1 


Add. 


Dif. of 
Hts. 


Dednct. 


Dif. of 
Hts. 


Deduct. 


.5879 


31 


48.3657 


1 


.0108 


31 


10.3811 


2 


1.2407 


32 


50.9629 


2 


.0432 


32 


11.0617 


8 


1.9583 


33 


A3.6250 


8 


.0972 


38 


11.7638 


4 


2.7407 


34 


56.3618 


4 


.1728 


84 


12.4876 


6 


3.5879 


35 


50.1436 


5 


.2700 


36 


13.2330 


6 


45000 


36 


62.0000 


6 


.8889 


86 


14.0000 


7 


5.4768 


37 


64.9218 


7 


.6293 


37 


147885 


8 


6.5185 


38 


67.9073 


8 


.6913 


88 


15.6987 


9 


7.6260 


39 


70.9583 


9 


.8760 


39 


16.4306 


10 


8.79^ 


40 


74.0740 


10 


1.0802 


40 


17,2889 


11 


10.0324 


41 


77.2456 


11 


1.3071 


41 


18.1589 


12 


11.3333 


42 


80.6000 


12 


1.6565 


42 


19.0666 


13 


12.6990 


43 


83.8101 


13 


1.8256 


48 


19.9747 


14 


14.1296 


44 


87.1852 


14 


2.1173 


44 


20.9136 


16 


15.6250 


45 


90.6250 


15 


2.4306 


45 


21.8760 


16 


17.1852 


46 


941296 


16 


2.7654 


46 


22.8580 


17 


18.8101 


47 


97.6990 


17 


3.1219 


47 


23.8626 


18 


20.6000 


48 


101.3333 


18 


3.5000 


48 


248888 


19 


23.2546 


49 


105.0324 


19 


8.8997 


49 


25.9367 


20 


24.0740 


50 


108.7962 


20 


43210 


50 


27.0061 


21 


25.9683 


51 


112.6260 


21 


4.7689 


51 


28.0972 


22 


27.9073 


52 


116.6186 


22 


5.2284 


52 


29.2098 


23 


29.9213 


53 


120.4768 


23 


5.7145 


53 


30.3441 


24 


32.0000 


64 


124,600J 


24 


6.2222 


64 


31.5000 


25 


34.1435 


65 


128.5879 


25 


6.7616 


55 


32.6774 


26 


36.3518 


56 


132.7407 


26 


7.3024 


66 


33.8765 


27 


38.6260 


67 


186.9583 


27 


7.8750 


57 


35.0972 


28 


40.9629 


58 


141.2407 


28 


8.4691 


58 


86.8894 


29 


43.3657 


69 


146.6879 


29 


9.0648 


69 


37.6033 


30 


45.8333 


60 


150.0000 


30 


9.7222 


60 188.8888 



\ 



( cviii. ) 


BA8B 80— SLOPE 2 to 1. 


1 

» 


Add. 


1, 
•s 

ta 


Add. 


1 


Deduct. 




Dednot. 


1 


.5926 


31 


52.8148 


.0123 


31 


11.8642 


2 


1.2592 


82 


66.7037 


2 


.0494 


32 


12.6419 


8 


2.0000 


88 


68.6666 


8 


.1111 


33 


13.4444 


4 


2.8148 


34 


61.7037 


4 


.1975 


34 


14.2716 


6 


3.7087 


86 


64.8148 


6 


.3086 


36 


16.1234 


6 


4.6666 


3B 


68.0000 


e 


.4444 


36 


16.0000 


7 


6.7037 


37 


71.2592 


7 


.6049 


87 


1^9012 


8 


6.8148 


88 


74.5926 


8 


.7901 


88 


17.8271 


9 


8.0000 


39 


78.0000 


9 


1.0000 


39 


18.7777 


10 


9.2592 


40 


81.4814 


10 


1.2346 


40 


19.7630 


11 


10.5926 


41 


85.0870 


11 


1.4938 


41 


20.7530 


12 


12.0000 


42 


88.6666 


12 


1.7778 


42 


21.7777 


13 


13.4814 


43 


92.3074 


13 


2.0864 


48 


22.8271 


14 


15.0370 


44 


96.1480 


14 


2.4197 


44 


23.9012 


15 


1&6666 


46 


100.0000 


16 


2.7778 


46 


26.0000 


16 


18.3704 


46 


103.9269 


16 


ai606 


46 


26.1234 


17 


20.1480 


47 


107.9259 


17 


3.5679 


47 


27.2716 


18 


22.0000 


48 


112.0000 


18 


4.0000 


48 


28.4444 


19 


24.9259 


49 


116.1480 


19 


4.4568 


49 


29.6^20 


20 


25.9259 


60 


120.3704 


20 


4.9882 


50 


80.8642 


21 


28.0000 


61 


124.6666 


21 


6.4444 


51 


82.1111 


22 


30.1480 


52 


129.0870 


22 


6.9768 


52 


33.3827 


23 


32.3704 


63 


133.4814 


28 


a5809 


58 


34.6790 


24 


34.6666 


64 


138.0000 


24 


7.1111 


54 


86.0000 


25 


37.0370 


66 


1^.5920 


26 


7.7160 


65 


37.3456 


26 


89.4814 


66 


147.2592 


26 


8.3457 


66 


38.7160 


27 


42.0000 


67 


152.0000 


27 


9.0000 


57 


40.1111 


28 


44.5926 


58 


156.8148 


28 


9.6790 


58 


41.5308 


29 


47.2692 


59 


161.7037 


29 


10.8827 


S9 


42.9763 


80 


50.0000 601 


166.6666 


80 11.1111 1 60 


44.4444 







( c«- ) 






BASE 30-SLOPE 2^ to 1. 


€ 




•a 




"Si 




^i 




3 


Add. 


S 


Add. 


Urn 


Dednct. 


gas 


Dednet. 


1 


.6018 


31 


61.7130 


1 


.0154 


31 


148302 


2 


1.29^ 


32 


65.1651 


2 


.0617 


% 


15.8016 


3 


2.0833 


33 


68.7500 


3 


.1389 


33 


16.8056 


4 


2.9630 


34 


72.4074 


4 


.2468 


34 


17.8395 


5 


8.9352 


35 


76.1673 


6 


.3867 


36 


18.9043 


6 


5.0000 


36 


80.0000 


6 


.5566 


86 


20.0000 


7 


ai573 


37 


83.9352 


7 


.7561 


37 


21.1266 


8 


7.4074 


38 


87.9630 


8 


.9876 


38 


29,.2839 


9 


a7600 


39 


92.0638 


9 


1.2600 


39 


23.4722 


10 


10.1851 


40 


96.2962 


10 


1.6432 


40 


24.6918 


11 


11.7130 


41 


100.6018 


11 


1.8673 


41 


25.9413 


12 


13.3333 


42 


105.0000 


12 


2.2222 


42 


27.2222 


13 


15.0462 


43 


109.4908 


13 


2.6060 


43 


28.6360 


14 


1&8518 


44 


114.0740 


14 


3.0247 


44 


29.8765 


15 


18.7500 


45 


118.75(X) 


15 


a4722 


46 


31.2600 


16 


20.7407 


46 


123.5185 


16 


a9506 


46 


32.6643 


17 


22.8240 


47 


128.3797 


17 


44599 


47 


34.0895 


18 


25.0000 


48 


lS3.a333 


18 


5.0000 


48 


36.5665 


19 


28.2665 


49 


138.3797 


19 


5.5710 


49 


87.0524 


20 


29.6296 


50 


1445185 


20 


ai728 


50 


38.5802 


21 


32.0833 


51 


148.7600 


21 


a8056 


51 


40.1388 


22 


34.6296 


52 


154.0740 


22 


7.4691 


52 


41.7283 


23 


37.2686 


53 


159.4908 


23 


8.1636 


53 


43.3487 


24 


40,0000 


54 


166.0000 


24 


8.8889 


54 


46.0000 


25 


42.8240 


55 


170.6018 


26 


9.6465 


56 


4&6820 


26 


45.7407 


56 


178.2962 


26 


10.4321 


56 


48.8960 


27 


48.7500 


67 


182.0833 


27 


11.2500 


67 


60.1388 


28 


51.8518 


56 


187.9680 


28 


12.0687 


58 


51.9185 


29. 


55.0462 


59 


193.9352 


29 


12.9782 


59 


53.7191 


30 


58.3333 


60 


200.0000 


30 


13.8889 


60 


65.5666 





(ex.) 




BASE SO— SLOPE 3 to 1. 


} 

as 


Add. 


1 


Add. 


IKf.of 
Hts. 


Deduct. 


Htt. 


Deduct. 


1 


.6111 


31 


70.6111 


1 


.0186 


31 


17.7963 


2 


1.3333 


32 


74.6666 


2 


.0740 


32 


ia9630 


3 


2.1606 


33 


78.8338 


3 


.1667 


38 


20.1666 


4 


8.1111 


34 


88.1111 


4 


.2963 


34 


21.4074 


5 


41666 


36 


87.6000 


6 


.4628 


85 


22.6861 


6 


6.3333 


36 


92.0000 


6 


.6667 


86 


24.0000 


7 


6.6111 


37 


96.6111 


7 


.9074 


87 


26.3518 


8 


8.0000 


88 


101.3383 


8 


1.1852 


88 


26.7407 


9 


9.6000 


39 


106.1666 


9 


1.5000 


39 


28.1666 


10 


11.1111 


40 


111.1111 


10 


1.8518 


40 


29.6296 


11 


12.8333 


41 


116.1666 


11 


2.2407 


41 


31.1296 


12 


14.6666 


42 


121.3838 


12 


2.6667 


42 


32.6666 


13 


16.6111 


48 


126.6111 


13 


8.1296 


48 


84.2407 


14 


18.6666 


44 


132.0000 


14 


3.6296 


44 


36.8618 


16 


20.8333 


46 


137.6000 


16 


4.1667 


45 


37.5000 


16 


23.1111 


46 


143.1111 


16 


4.7407 


46 


39.1851 


17 


25.6000 


47 


149.8333 


17 


6.3518 


47 


40.9078 


18 


28.0000 


48 


164.6666 


18 


6.0000 


48 


42.6666 


19 


31.6111 


49 


160.6111 


19 


6.6862 


49 


444629 


20 


83.3333 


60 


167.6666 


20 


7.4074 


50 


46.2963 


21 


36.1666 


61 


172.8338 


21 


8.1667 


61 


49.1666 


22 


39.1111 


62 


179.1111 


22 


8.9629 


62 


50.0740 


23 


42.1666 


68 


186.6000 


28 


9.7962 


63 


62.0186 


24 


46.3833 


64 


192.0000 


24 


10.6667 


64 


540000 


26 


48.6111 


66 


198.6111 


26 


11.5741 


56 


66.0184 


26 


62.0000 


66 


206.38.33 


26 


12.5184 


56 


68.0740 


27 


66.6000 


57 


212.1666 


27 


13.5000 


67 


60.1666 


28 


60.1111 


58 


219.1111 


28 


14.6186 


68 


62.2962 


29 


62.8333 


69 


226.1666 


29 


15.6741 


59 


644620 


30 


66.6666 


60 


233.8333 


30 


1&6667 60 


66.6666 





( c"- ) 




BASE 31— SLOPE 


; 1 to 1. 


M Height 


Add. 


t 


Add. 


1 


Deduct 




Deduct. 


.6787 


31 


22.2463 


.0016 


31 


1.4830 


2 


1.1666 


32 


23.1111 


2 


.0062 


32 


1.6802 


3 


1.7689 


33 


23.9661 


3 


.0139 


33 


1.6805 


4 


2.3703 


34 


248703 


4 


.0246 


34 


1.7839 


6 


2.9661 


36 


26.7639 


6 


.0385 


35 


1.8904 


6 


aeiii 


36 


26.6666 


6 


.0555 


36 


2.0000 


7 


4.2453 


87 


27.6787 


7 


.0756 


37 


2.1126 


8 


4.8888 


38 


28.5000 


8 


•UUOo 


38 


2.^84 


9 


6M16 


39 


29.4305 


9 


.1250 


39 


2.3472 


10 


6.2037 


40 


30.3703 


10 


.1643 


40 


2.4691 


11 


6.8760 


41 


31.3194 


11 


.1867 


41 


2.6941 


12 


7.5555 


^ 


32.2777 


12 


.2222 


42 


2.7222 


13 


8.2453 


43 


33.2463 


13 


.2608 


43 


2.8646 


14 


8.9444 


44 


34.2222 


14 


.3026 


44 


2.9876 


15 


9.6627 


46 


35.2083 


15 


.3472 


45 


3.1250 


16 


10.3703 


46 


36.2037 


16 


.3951 


46 


3.2664 


17 


11.0972 


47 


37.2083 


17 


.4460 


47 


3.4089 


18 


11.8333 


48 


38.2222 


18 


.6000 


48 


3.6566 


19 


12.5787 


49 


39.2463 


19 


.5571 


49 


3.7052 


20 


13.3333 


60 


40.2777 


20 


.6173 


60 


3.8580 


21 


14.0972 


61 


41.3194 


21 


.6805 


61 


40139 


22 


14.8703 


52 


42.3703 


22 


.7469 


52 


41728 


23 


16.6527 


63 


43.4305 


23 


.8163 


53 


43349 


24 


16.4444 


64 


44.6000 


24 


.8889 


64 


4.6000 


25 


17.2453 


55 


46.5787 


25 


.9647 


65 


46682 


26 


18.0556 


56 


46.6666 


26 


1.0432 


66 


48396 


27 


18.8750 


67 


47.7639 


27 


1.1250 


67 


5.0139 


28 


19.7037 


68 


48.8703 


28 


1.2099 


58 


5.1913 


29 


20.5416 


59 


49.9861 


29 


1.2978 


69 


6.3719 


30 


21.3888 


60 


51.1111 


30 


1.3889 


60 


5.6656 



< 




( cxii. ) 




BASE 31— SLOPE 

• 


; ^ to 1. 


} 


Add. 


1 


Add. 


if. of 
Eits. 


Dednot. 


3^ 


Deduct. 


» 




» 




1 




&^ 




1 


.5833 


31 


26.6044 


.0031 


31 


2.9660 


2 


1.1851 


32 


27.8518 


2 


.0123 


32 


3.3611 


9 


1.8055 


33 


29.0277 


8 


.0278 


88 


3.6111 


4 


2.4444 


34 


30.2222 


4 


.0494 


34 


3.5679 


5 


8.1018 


35 


31.4862 


5 


.0772 


36 


3.7808 


6 


3.7777 


36 


32.6666 


6 


.1111 


36 


4.0000 


7 


4.4722 


87 


33.9166 


7 


.1512 


tn 


4.2253 


8 


5.1851 


38 


36.1861 


8 


.1976 


88 


4.4568 


9 


5.9166 


89 


36.4722 


9 


.2500 


89 


4.0944 


10 


6.6666 


40 


37.7777 


10 


•OUCtU 


40 


4.9383 


11 


7.4352 


41 


39.1018 


11 


.8734 


41 


5.1883 


12 


8.2222 


42 


40.4444 


12 


.4444 


42 


5.4444 


13 


9.0277 


43 


41.8065 


13 


.6216 


43 


6.7076 


14 


9.8518 


44 


43.1851 


14 


.6049 


44 


5.9753 


15 


10.6944 


45 


44.6633 


15 


.6944 


45 


6.2500 


16 


11.5665 


46 


46.0000 


16 


.7901 


46 


a5308 


17 


12.4362 


47 


47.4351 


17 


.8920 


47 


6.8179 


18 


13.3333 


48 


48.8888 


18 


1.0000 


48 


7.1111 


19 


14.2600 


49 


50.3611 


19 


1.11^ 


49 


7.4106 


20 


16.1851 


50 


51.8518 


20 


1.2346 


60 


7.7160 


21 


16.1388 


61 


53.3611 


21 


1.3611 


61 


8.0277 


22 


17.1111 


52 


64.8888 


22 


1.4088 


62 


8.3456 


23 


18.1018 


53 


66.4351 


23 


1.6327 


53 


8.6697 


24 


19.1111 


54 


68.0000 


24 


1.7778 


64 


9.0000 


25 


20.1388 


56 


59.5838 


25 


1.9295 


56 


9.3364 


26 


21.1851 


56 


61.1861 


26 


2.0864 


66 


9.6790 


27 


22.2600 


57 


62.8055 


27 


2.2600 


57 


10.0277 


28 


23.3333 


58 


64.4444 


28 


2.4197 


58 


10.3827 


29 


24.4362 


59 


66.1018 


29 


2.6966 


69 


10.7438 


30 


26.5655 


60 


67.7777 


30 


2.7778160 


11.1111 





( cxiii. ) 


BASE 31— SLOPE f to 1. 


1 
.-e 


Add. 


1 


Add. 


"Si 


Dednet. 


2^ 

■sn 


Deduct. 


IS 




a 




o 




ft 




1 


.5880 


31 


31.1436 


1 


.0046 


31 


44490 


2 


1.2037 


32 


32.5926 


2 


.0186 


32 


47407 


8 


1.8472 


33 


34.0694 


3 


.0416 


33 


6.0416 


4 


2.5185 


34 


35.5740 


4 


.0740 


34 


5.8618 


5 


3.2176 


36 


37.1066 


5 


.1167 


36 


6.6712 


6 


a9444 


36 


38.6666 


6 


.1667 


36 


6.0000 


7 


4.6990 


37 


40.2646 


7 


.2268 


37 


6.3379 


8 


5.4814 


38 


41.8703 


8 


.2963 


38 


6.6861 


9 


a2916 


39 


43.6140 


9 


.8750 


39 


7.0416 


10 


7.1296 


40 


46.1862 


10 


.4630 


40 


7.4074 


11 


7.9954 


41 


46.8842 


11 


.5602 


41 


7.7824 


12 


8.8888 


42 


48.6111 


12 


.6667 


42 


8.1666 


13 


9.8102 


43 


50.3667 


13 


.7824 


43 


8.5612 


14 


10.7692 


44 


62.1481 


14 


.9074 


44 


8.9629 


15 


11.7360 


46 


63.9583 


15 


1.0417 


46 


9.3750 


16 


12.7407 


46 


56.7963 


16 


1.1852 


46 


9.7962 


17 


13.7732 


47 


57.6620 


17 


1.3379 


47 


10.2268 


18 


14.8333 


48 


69.6655 


18 


1.6000 


48 


10.6666 


19 


16.9213 


49 


61.4768 


19 


1.6713 


^ 


11.1157 


20 


17.0370 


60 


63.4260 


20 


1.8518 


50 


11.6740 


21 


18.1805 


61 


66.4028 


21 


2.0417 


51 


12.0416 


22 


19.3518 


62 


67.4074 


22 


2.2407 


62 


12.5185 


23 


20.6509 


63 


69.4397 


23 


2.4491 


53 


13.0046 


24 


21.7777 


64 


71i»00 


24 


2.6667 


54 


13.6000 


25 


23.0323 


55 


73.6879 


26 


2.8935 


55 


140046 


26 


24.3147 


56 


76.7037 


26 


3.1296 


56 


146185 


27 


25.6250 


57 


77.8472 


27 


3.3760 


57 


15.0416 


28 


26.9629 


68 


80.0185 


28 


3.6296 


68 


15.5740 


29 


28.3288 


59 


82.2175 


29 


3.8936 


59 


16.1167 


30 


29.72S» 


60 


844444 


30 1 4.1667 


60 16.6666 





(cxiv. ) 




BASE 31— SLOPE 1 to 1. 


1 


Add. 


1 


Add. 


Dif.of 
Hts. 


Deduct 


Dif.of 


Deduct. 


.5926 


31 


35.6926 


1 


.0062 


81 


5.9321 


2 


1.2222 


32 


37.3333 


2 


.0247 


82 


6.3210 


3 


1.8888 


S3 


39.1111 


3 


.0566 


83 


6.7222 


4 


2.5926 


34 


40.9260 


4 


.0968 


34 


7.1368 


5 


3.3333 


85 


. 42.7777 


5 


.1643 


35 


7.5617 


6 


4.1111 


86 


446666 


6 


.2222 


86 


8.0000 


7 


49260 


87 


46.6926 


7 


.3025 


87 


8.4606 


8 


6.7777 


38 


48.5566 


8 


.8961 


88 


8.9135 


9 


6.6666 


89 


60.6656 


9 


.6000 


89 


9.3888 


10 


7.6926 


40 


52.5926 


10 


.6178 


40 


9.8766 


11 


8.5556 


41 


646666 


11 


.7460 


41 


10.3766 


12 


9.6655 


42 


66.7777 


12 


.8889 


42 


10.8888 


13 


10.5926 


43 


58.9260 


13 


1.0432 


43 


11.4136 


14 


11.6666 


44 


61.1111 


14 


1.2099 


44 


11.9506 


16 


12.7777 


46 


63.3333 


16 


1.3889 


45 


12.6000 


16 


13.9260 


46 


65.6026 


16 


1.5802 


46 


ia0617 


17 


16.1111 


47 


67.8888 


17 


1.7889 


47 


13.6868 


18 


1&3333 


48 


70.2222 


18 


2.0000 


48 


142222 


19 


17.6926 


49 


72.6926 


19 


2.2284 


49 


148209 


20 


18.8888 


50 


75.0000 


20 


2.4691 


60 


15.4321 


21 


20.2222 


51 


77.4444 


21 


2.7222 


61 


16.0555 


22 


21.6»iM 


m 


79.9260 


22 


2.9876 


52 


16.6918 


23 


23.0000 


63 


82.4444 


23 


3.2664 


63 


17.3396 


24 


244444 


64 


86.0000 


24 


8.6655 


54 


18.0000 


25 


26.9260 


66 


87.6926 


25 


3.8630 


65 


18.6728 


26 


27.4444 


66 


90.2222 


26 


41728 


56 


19.3560 


27 


29.0000 


57 


92.8888 


27 


45000 


ff7 


20.0565 


28 


30.5926 


58 


96.5926 


28 


48306 


58 


20.7664 


20 


32.2222 


69 


98.8338 


29 


5.1913 


69 


21.4876 


30 


33.8888 


60 


101.1111 


30 


5.5665 


60 





( <a^- ) 


BASE 31— SLOPE IJ to 1. 


1 


Add. 


1 


Add. 


if. of 
Hta. 


Dednct 


3^ 


Dedsct. 


s 




!S 




o 






- 


1 


.S972 


31 


40.0416 


1 


.0077 


31 


7.4161 


2 


1.2407 


32 


42.0740 


2 


.0809 


82 


7.9012 


8 


1.9305 


83 


44.1528 


3 


.0694 


.33 


8.4027 


4 


2.6666 


34 


46.2777 


4 


.1234 


84 


a9197 


5 


a4490 


86 


48.4490 


5 


.1928 


35 


9.4621 


6 


4.2777 


86 


50.6666 


6 


.2778 


36 


10.0000 


7 


5.1538 


37 


52.^05 


7 


.8781 


37 


10.6632 


8 


a0740 


38 


55.2407 


8 


.'^38 


38 


11.1419 


9 


7.0416 


39 


57.6972 


9 


.6260 


39 


11.7361 


10 


8.0555 


40 


eaoooo 


10 


.7716 


40 


12.8466 


11 


9.1167 


41 


62.4490 


11 


.9336 


41 


12.9706 


12 


10.2222 


ASl 


64.9444 


12 


1.1111 


42 


13.6111 


13 


1L8760 


43 


67.4862 


13 


1.3040 


43 


14.2680 


14 


12.5740 


44 


70.0740 


14 


1.6123 


44 


14.9382 


15 


13.8103 


45 


72.7083 


16 


1.7861 


45 


16.6250 


16 


15.1111 


46 


76.3888 


16 


1.9763 


46 


16.3271 


17 


16.4490 


47 


78.1166 


17 


2.2299 


47 


17.0447 


18 


17.8333 


48 


80.8888 


18 


2.5000 


48 


17.7T77 


19 


19.2640 


49 


88.7088 


19 


2.7866 


49 


18.6262 


20 


20.7407 


50 


86.5740 


20 


8.0864 


60 


19.2901 


21 


22.2640 


61 


89.4862 


'21 


8.4028 


61 


20.0694 


22 


23.8333 


52 


92.4444 


22 


3.7846 


62 


20.8641 


23 


25.4490 


53 


95.4490 


23 


4.0818 


63 


21.6743 


24 


27.1111 


54 


98.6000 


24 


4.4444 


64 


22.5000 


26 


28.8193 


65 


101.6972 


25 


4.8228 


66 


23.8410 


26 


30.5740 


56 


104.7407 


26 


5.2166 


66 


241975 


37 


32.8760 


67 


107.9305 


27 


6.6260 


57 


25.0694 


28 


34.2222 


68 


111.1666 


28 


6.0^4 


68 


25.9667 


29 


36.1167 


69 


1144^0 


29 


6.4891 


69 


26.8696 


30 


38.0555 


60 


117.7777 


30 


6.9444 


60 


27.7777 





(cxvu) 


BASE 81--SL0PE IJ to 1. 


2 




£ 




"8 J 




% . 




B 
1 


Add. 


» 


Add. 


^1 


Dedncb 


g« 


Dedont. 


.6018 


31 


44.^07 


1 


.0092 


31 


8.8981 


2 


1.2692 


32 


46.8148 


2 


.0370 


32 


9.4815 


3 


1.975» 


33 


49.1944 


8 


.0633 


33 


10.0883 


4 


2.7407 


34 


61.6296 


4 


.1480 


34 


10.7037 


6 


3i!647 


36 


541203 


5 


.2313 


35 


11.3^5 


6 


AAAAA 


36 


56.6666 


6 


.3383 


86 


12.0000 


7 


6.3796 


37 


59.2684 


7 


.4637 


37 


12.6769 


8 


ft8703 


38 


61.9259 


8 


.6926 


38 


13.8708 


9 


7.4166 


39 


646388 


9 


.7500 


39 


140833 


10 


8i»186 


40 


67.4074 


10 


.9259 


40 


148148 


11 


9.6769 


41 


70.2314 


11 


1.1208 


41 


15M48 


12 


10.8888 


42 


73.1111 


12 


1.8333 


42 


1&8333 


13 


12.1674 


43 


76.0462 


13 


1.5648 


43 


17.1202 


14 


13.4816 


44 


79.0370 


14 


1.8148 


44 


17.9269 


15 


148611 


46 


82.0833 


15 


2.0633 


45 


18.7500 


16 


16.2962 


46 


85.1861 


16 


2.3704 


46 


19.5926 


17 


17.7870 


47 


88.3424 


17 


2.6769 


47 


20.4587 


18 


19.3333 


48 


91.6655 


18 


3.0000 


48 


21.8383 


19 


20.0360 


49 


948240 


19 


ad426 


^ 


22.2314 


20 


22.6926 


50 


98.1482 


20 


a7087 


50 


23.1481 


21 


24.3056 


61 


101.5277 


21 


40888 


61 


240833 


22 


26.0740 


52 


1049629 


22 


4.4815 


62 


26.0370 


23 


27.8982 


53 


106.4636 


23 


48981 


63 


26.0092 


24 


29.7777 


54 


112.0000 


24 


5.3333 


64 


27.0000 


25 


31.7130 


65 


115.6018 


26 


5.7869 


56 


28.0092 


26 


33.7037 


56 


119.2692 


26 


6.2692 


56 


29.0370 


27 


36.7600 


57 


122.97^ 


27 


6.7500 


57 


30.0833 


28 


37.8618 


68 


126.7407 


28 


7.2592 


58 


81.1481 


29 


40.0092 


69 


130i>647 


29 


7.7869 


59 


32.2314 


80 


42.2222 60 


1344444 


30 


8.8333 


60 


88.8383 





( cxvii. ) 




BASE 81— SLOPE If to 1. 


1 

m 


Add. 


1 


Add. 


Dif.of 
Hts. 


Deduct. 


Dif.of 
Hts. 


Deduct. 


1 


.6064 


31 


48.9998 


1 


.0108 


31 


10.8811 


2 


1.2777 


32 


61.5665 


2 


.0432 


32 


11.0617 


3 


2.0189 


33 


542860 


3 


.0972 


88 


11.7638 


4 


2.8148 


34 


5&9815 


4 


.1728 


84 


12.4876 


5 


a6804 


35 


59.7916 


5 


.2700 


85 


18.2330 


6 


4.6111 


36 


62.6666 


6 


.3889 


36 


140000 


7 


5.6064 


87 


65.6064 


7 


.5293 


37 


147885 


8 


6.6666 


88 


68.6111 


8 


.6913 


38 


16.5987 


9 


7.7916 


89 


71.6804 


9 


.8760 


39 


16.4305 


10 


8.9815 


40 


74.8148 


10 


1.0802 


40 


17.2839 


11 


10.2360 


41 


78.0139 


11 


1.3071 


41 


18.1689 


12 


11.5556 


^ 


81.2777 


12 


1.5566 


^ 


19.0565 


18 


12.9398 


43 


84.6064 


13 


1.8266 


43 


19.9747 


14 


143888 


44 


88.0000 


14 


2.1173 


44 


20.9185 


Id 


15.9027 


46 


91.4583 


15 


2.4306 


45 


21.8760 


16 


17.4815 


46 


94.9814 


16 


2.7664 


46 


22.8580 


17 


19.1260 


47 


98.5694 


17 


8.1219 


47 


23.8626 


18 


20.8333 


48 


102.2222 


18 


3.5000 


48 


24.8888 


19 


22.6064 


49 


105.9397 


19 


3.8997 


49 


25.9367 


20 


24.4444 


60 


109.7222 


20 


4.3210 


50 


27.0061 


21 


2&8470 


61 


118.5694 


21 


47639 


61 


28.0972 


22 


28.3147 


62 


117.4814 


22 


5.2284 


52 


29.2098 


28 


30.3470 


58 


121.4683 


23 


6.7146 


53 


30.8441 


24 


32.4444 


64 


126.5000 


24 


6.2222 


64 


81.5000 


25 


34.6064 


56 


129.6064 


25 


a7616 


55 


82.6774 


26 


d&8833 


56 


133.7777 


26 


7.3024 


66 


33.8766 


27 


89.1250 


67 


138.0140 


27 


7.8760 


67 


35.0972 


28 


41.4815 


58 


142.3148 


28 


8.4691 


68 


86.3394 


29 


43.9027 


59 


146.6804 


29 


9.0848 


59 


87.6033 


80 


46.8888 


60 


161.1111 


80 


9.7222 


60 


38.8888 





■■; " " ■ ' - .-_,T., . ■ . - 
( cxTiii. ) 


BASE 31— SLOPE 2 to 1. 


i 

m 
1 


Add. 


1 

m 


Add. 


Dif.of 
Hts. 


Deduct 


Dif.of 
Uts. 


Deduct 


.6111 


31 


53.3888 


1 


.0123 


81 


11.8642 


2 


1.2962 


82 


66.2962 


2 


.0494 


32 


12.6419 


8 


2.0556 


33 


69.2777 


8 


.1111 


33 


ia4444 


4 


2.8888 


34 


62.8838 


4 


.1976 


84 


142716 


5 


a7962 


36 


65.4629 


5 


.3086 


36 


15.1234 


6 


4.7777 


86 


68.6666 


6 


.4444 


36 


16.0000 


7 


5.8383 


37 


71.9444 


7 


.6049 


37 


16.9012 


8 


6.9629 


88 


76.2962 


8 


.7901 


38 


17.8271 


9 


8.1666 


39 


78.7222 


9 


1.0000 


39 


18.7777 


10 


9.4444 


40 


82.2222 


10 


1.2346 


40 


19.7580 


11 


10.7962 


41 


85.7962 


11 


1.^38 


41 


20.7580 


12 


12.2222 


^ 


89.4444 


12 


1.7778 


42 


21.7777 


13 


13.7222 


48 


93.1666 


13 


2.0864 


43 


22.8271 


14 


15.2962 


44 


96.9^29 


14 


2.4197 


44 


23.9012 


15 


16.9444 


46 


100.8338 


16 


2.7778 


45 


26.0000 


16 


18.6666 


46 


1047777 


16 


3.1605 


46 


26.1234 


17 


20.4629 


47 


108.7962 


17 


3.5679 


47 


27.2716 


18 


22.3333 


48 


112.8888 


18 


40000 


48 


28.4444 


19 


242777 


^ 


117.0656 


19 


44668 


49 


29.6420 


20 


26.2962 


50 


121.2962 


20 


49382 


60 


30.86^ 


21 


2ad888 


51 


126.6111 


21 


6.4444 


51 


82.1111 


22 


30.5656 


62 


130.0000 


22 


6.9758 


52 


33.8827 


23 


32.7962 


68 


1344629 


28 


6.5309 


53 


84.6790 


24 


35.1111 


64 


139.0000 


24 


7.1111 


54 


36.0000 


26 


37.6000 


55 


148.6111 


26 


7.7160 


56 


37.8466 


26 


39.9629 


66 


14a2962 


26 


8.8467 


66 


38.7160 


27 


42.6000 


67 


163.0556 


27 


9.0000 


67 


40.1111 


28 


45.1111 


58 


167.8888 


28 


9.6790 


58 


41.6308 


29 


47.7962 


69 


162.7962 


29 


10.8827 


69 


^9758 


80 160.5555 


60 


167.7777 


80 


11.1111 


60 


44.4444 



( cziz. ) 


BASE 31— SLOPE 2( to 1. 


1 

-8 


Add. 


1 


Add. 


if. of 
Hti. 


Dedaet. 


if. of 
Hti. 


Deduct. 


a 




m 




o 




P 




1 


.6208 


31 


62.2870 


1 


.0154 


81 


148302 


2 


1.3883 


82 


65.7777 


2 


.0617 


32 


16.8013 


3 


2.1388 


33 


69.8611 


3 


.1889 


33 


16.8056 


4 


a0370 


34 


73.0870 


4 


.2468 


84 


17.8895 


5 


4.0277 


35 


7&8055 


5 


.3857 


35 


18.9048 


6 


5.1111 


36 


c5U.tJduD 


6 


.5555 


36 


20.0000 


7 


&2870 


37 


84.6202 


7 


.7561 


87 


21.1265 


8 


7.55S5 


38 


oo.UIdo 


8 


.9676 


38 


22.2839 


9 


8.9166 


39 


92.8055 





1.2500 


39 


28.4722 


10 


10.8708 


40 


97.0370 


10 


1.54% 


40 


246918 


11 


11.9166 


41 


101.8611 


11 


1.8673 


41 


25.9413 


12 


ia5555 


^ 


105.7777 


12 


2.2222 


4@ 


27.2222 


13 


15.2870 


43 


110.2870 


13 


2.6080 


48 


28.6860 


14 


17.1111 


44 


1148888 


14 


3.0247 


44 


29.8765 


15 


19.0277 


45 


119.5883 


15 


3.4722 


45 


31.2500 


16 


21.0870 


46 


124.3708 


16 


8.9506 


46 


32.6643 


17 


23.1388 


47 


129.2500 


17 


44599 


47 


840896 


18 125.3833 

19 127.6208 


48 


1342222 


18 


5.0000 


48 


35.5665 


49 


139.2870 


19 


5.6710 


49 


37.0524 


20 


30.0000 


50 


1444444 


20 


6.1728 


60 


38.6802 


21 


34.4722 


51 


149.6944 


21 


6.8055 


51 


40.1388 


22 


35.0370 


62 


155.0870 


22 


7.4691 


52 


41.7283 


23 


37.6944 


63 


160.4722 


23 


8.1686 


63 


43.3487 


24 


40.4444 


54 


166.0000 


24 


8.8889 


54 


45.0000 


25 


43.2870 


55 


171.6208 


25 


9.6455 


56 


46.6820 


26 


46.2St22 


56 


177.8338 


26 


10.4321 


66 


48.3950 


27 


^.2500 


57 


183.1388 


27 


11.2500 


57 


50.1888 


28 


62.3708 


58 


189.0370 


28 


12.0587 


68 


51.9136 


29 


55.5833 


69 


195.0277 


29 


12.9782 59 


53.7191 


ao 


58.8888 


60 


201.1111 


80 


18.8889 60 65.6555 



( c«- ) 




BASE 81— SLOPE 3 to 1. 




:c 


Add. 


i 

a 


Add. 


1 


Dodnct. 


■Si 


Deduct. 




1 


.6296 


31 


71.1851 


.0185 


31 


17.7963 




2 


1.3704 


32 


75.2592 


2 


.0740 


32 


18.9630 




3 


2.2222 


33 


79.4444 


3 


.1667 


33 


20.1666 




4 


3.1851 


34 


83.7407 


4 


.2963 


34 


21.4074 




6 


4.2592 


35 


88.1481 


6 


.4628 


35 


22.6851 




6 


5.4444 


36 


92.6066 


6 


.6667 


36 1240000 1 




7 


a7407 


37 


97.2962 


7 


.9074 


87 


25.3518 




8 


8.1481 


38 


102.0370 


8 


1.1852 


38 


2&7407 




9 


9.6666 


39 


106.8888 


9 


1.5000 


39 


28.1666 




10 


11.2962 


40 


111.8518 


10 


1.8518 


40 


29.6296 




11 


13.0370 


41 


116.9260 


11 


2.2407 


41 


31.1296 




12 


148888 


42 


122.1111 


12 


2.6667 


42 


32.6666 




13 


16.8518 


43 


127.4074 


13 


3.1296 


43 


342407 




14 


18.9260 


44 


132.8146 


14 


3.6296 


44 


35.8518 




15 


21.1111 


45 


138.3333 


15 


41667 


45 


37.5000 




16 


23.4074 


46 


143.9629 


16 


47407 


46 


39.1851 




17 


25.8147 


47 


149.7037 


17 


5.3518 


47 


40.9074 




18 


28.3333 


48 


155.5555 


18 


6.0000 


48 


42.6666 




19 


30.9629 


4d 


161.5185 


19 


&6852 


49 


44.4629 




20 


33.7037 


50 


167.5926 


20 


7.4074 


50 


46.2963 




21 


36.5555 


51 


173.7777 


21 


8.1667 


51 


49.1666 




22 


39.5185 


52 


180.0740 


22 


8.9629 


52 


50.0740 




23 


42.5926 


53 


186.4814 


23 


9.7962 


53 


52.0185 




24 


45.7777 


54 


193.0000 


24 


10.6667 


54 


540000 




25 


49.0740 


55 


199.6296 


25 


11.5741 


55 


56.0184 




26 


52.4814 


56 


20&3703 


26 


12.5184 


56 


58.0740 




27 


56.0000 


57 


213.2222 


27 


13.5000 


67 


60.1666 




28 


59.6296 


58 


220.1852 


28 


145185 


58 


62.2962 




29 


63.3703 59 


227.2592 


29 


15.5741 59 1 


64.4629 




30 


67.2222 60 234.4444 


30 16.66671 60 f 


6&6666 




1 











( cxxi. ) 


BASE 32— SLOPE i to 1. 




Add. 


i 


Add. 


"A 


Deduct. 




1 

Deduct 


1 




X 




1 




31 




.5972 


81 


22-8194 


.0015 


1.4830 


2 


1.2037 


82 


23.7037 


2 


.0062 


82 


1.5802 


3 


1.8194 


33 


245972 


3 


.0139 


33 


1.6805 


4 


2.4444 


34 


25.5000 


4 


.0246 


34 


1.7839 


5 


3.0787 


35 


26.4120 


6 


.0886 


36 


1.8904 


6 


8.7-222 


36 


27.3333 


6 


.0556 


86 


2.0000 


7 


4.3750 


37 


28.2639 


7 


.0756 


37 


2.1126 

1 


8 


5.0370 


38 


29.2037 


8 


.0988 


88 


2.2284 


9 


6.7083 


39 


30.1527 


9 


.1250 


39 


2.3472 


10 


6.3888 


40 


81.1111 


10 


.1643 


40 


2.4691 


11 


7.0787 


41 


82.0787 


11 


.1867 


41 


2.5941 


12 


7.7777 


42 


38.0555 


12 


.2222 


42 


2.7222 


13 


8.4861 


43 


34.0416 


13 


.2608 


43 


2.8545 


14 


9.2037 


44 


36.0370 


14 


.3025 


44 


2.9876 


15 


9.9305 


45 


86.0416 


15 


.3472 


45 


3.1250 


16 


10.6666 


46 


87^0555 


16 


.3961 


46 


3.2654 


17 


11.4120 


47 


88.0787 


17 


.4460 


47 


3.4089 


18 


12.1666 


48 


39.1111 


18 


.6000 


48 


S.5555 


19 


12.9305 


49 


40.1527 


19 


.6571 


49 


3.7052 


20 


13.7037 


50 


41.2037 


20 


.6173 


60 


a8580 


21 


14.4861 


51 


42.2639 


21 


.6805 


51 


4.0139 


22 


15.2777 


52 


43.8333 


22 


.7469 


62 


4.1728 


23 


16.0787 


53 


44.4120 


23 


.8163 


63 


4.3349 


24 


16.8888 


54 


46.5000 


24 


.8889 


54 


4.5000 


25 


17.7083 


55 


46.6972 


26 


.9647 


65 


4.6682 


26 


18.5370 


56 


47.7037 


26 


1.0432 


56 


4.8395 


27 


19.3750 


67 


48.8194 


27 


1.1260 


57 


5.0139 


28 


20.2222 


68 


49.9444 


26 


1.2099 


58 


5.1913 


29 


21.0787 


69 


51.0787 


29 


1.2978 


69 


6.3719 


30 


21.9444 


60 


52.2222 


80 


1.8889 


60 


6.5555 


■ 


R 





( cxxii. ) 


BASE 3^-SLOPE ^ to 1. 


1 

1 


Add. 


1 

S 
31 


Add. 


I>if.of 


Dednet 




Deduct. 


.6018 


27.2685 


1 


.0031 


31 


2.9660 


2 


1.2222 


32 


28.4444 


2 


.0123 


32 


3.1605 


3 


1.8611 


33 


29.6389 


3 


.0278 


S3 


33611 


4 


2.6185 


34 


30.8618 


4 


.0494 


34 


3.5679 


5 


3.1944 


35 


32.0833 


6 


.0772 


35 


3.7808 


6 


a8888 


36 


33.3333 


6 


.1111 


86 


4.0000 


7 


4.6018 


37 


34.6018 


7 


.1612 


37 


4.2253 


8 


5.3333 


38 


36.8888 


8 


.1975 


38 


4.4568 


9 


6.0833 


39 


37.1944 


9 


.2500 


39 


4.6944 


10 


6.8518 


40 


38.6185 


10 


•oOoo 


40 


4.9383 


11 


7.6389 


41 


39.8611 


11 


.3734 


41 


5.1883 


12 


8.4444 


42 


41.2222 


12 


.4444 


42 


6.4444 


13 


9.2685 


43 


42.6018 


13 


.5216 


43 


6.7067 


14 


10.1111 


44 


44.0000 


14 


.6049 


44 


5.9763 


16 


10.9722 


46 


45.4166 


16 


.6944 


45 


6.2500 


16 


11.8518 


46 


46.8518 


16 


.7901 


46 


6.5308 


17 


12.7600 


47 


48.3a35 


17 


.8920 


47 


6.8179 


18 


13.6666 


48 


^.7777 


18 


1.0000 


48 


7.1111 


19 


14.6018 


49 


61.2684 


19 


1.1142 


49 


7.4106 


20 


15.5556 


60 


52.7777 


20 


1.2346 


50 


7.7160 


21 


16.5277 


51 


643056 


21 


1.3611 


51 


8.0277 


22 


17.6186 


62 


66.8518 


22 


1.4938 


62 


8.3466 


23 


18.5277 


63 


67.4166 


23 


1.6327 


63 


8.6607 


24 


19.5656 


64 


69.0000 


24 


1.7778 


64 


9.0000 


25 


20.6018 


55 


60.6018 


26 


1.9296 


55 


9.3364 


26 


21.6666 


66 


62.2222 


26 


2.0864 


66 


9.6790 


27 


22.7600 


57 


63.8611 


27 


2.2500 


57 


10.0277 


28 


23.8618 


58 


65.5185 


28 


2.4197 


68 


10.3827 


29 


24.9722 


59 


67.1944 


29 


2.5956 


59 


10.7438 


30 


2ailll 


60 


68.8888 


30 


2.7778 


60 


11.1111 



1 




( cxxiii. ) 


BASE 32— SLOPE | to 1. 


t 


Add. 


•t 


Add. 


"a 


Deduct. 


Jaaa 


Dednct. 


» 




a 

31 




0* 
1 




^^ p*^ 




1 


.6064 


31.7176 


.0046 


31 


4.4490 


2 


1.2407 


32 


33.1851 


2 


.0185 


32 


4.7407 


3 


1.9028 


33 


34.6805 


3 


.0416 


33 


5.0416 


4 


2.5926 


34 


36.2037 


4 


.0740 


34 


5.3618 


5 


3.3101 


35 


37.7846 


5 


.1157 


36 


66712 


6 


40555 


36 


39.3333 


6 


.1667 


86 


6.0000 


7 


4.8286 


37 


40.9397 


7 


.2268 


37 


63379 


8 


5.6296 


38 


42.5740 


8 


.2963 


38 


6.6851 


9 


6.4583 


39 


44.2360 


9 


.3750 


39 


7.0416 


10 


7.3148 


40 


45.9259 


10 


.4630 


40 


7.4074 


11 


8.1991 


41 


47.6435 


11 


.6602 


41 


7.7824 


12 


9.1111 


42 


49.3888 


12 


.6667 


42 


8.1666 


13 


10.0509 


43 


51.1620 


13 


.7824 


43 


8.6602 


14 


11.0185 


44 


52.9629 


14 


.9074 


44 


89629 


15 


12.0138 


45 


54.7916 


15 


1.0417 


46 


9.3750 


16 


13.0370 


46 


56.6481 


16 


1.1852 


46 


9.7962 


17 


14.0879 


47 


58.S323 


17 


1.3379 


47 


10.2268 


18 


15.1666 


48 


60.4444 


18 


1.5000 


48 


10.6666 


19 


16.2731 


49 


62.3842 


19 


1.6713 


49 


11.1157 


20 


17.4074 


50 


64.3518 


20 


1.8518 


50 


11.6740 


21 


18.5694 


51 


66.3472 


21 


2.0417 


51 


12.0416 


22 


19.7692 


52 


68.3703 


22 


2.2407 


62 


12.5186 


23 


20.9768 


53 


70.4212 


23 


2.4491 


53 


13.0046 


24 


22.2222 


54 


72.5000 


24 


2.6667 


64 


13.5000 


25 


23.4953 


55 


74.6064 


25 


2.8935 


65 


14.0046 


26 


24.7962 


56 


76.7407 


26 


3.1296 


66 


14.5185 


27 


26.1260 


57 


78.9028 


27 


8.3750 


57 


15.0416 


28 


27.4814 


58 


81.0926 


28 


3.6296 


68 


15.6740 


29 


28.8657 


59 


83.3101 


29 


3.8935 


59 


16.1157 


30 


30.2777 


60 


85.5555 


30 


4.1667 1 60 


lft6666 



1 


• 

( cxxiv. ) 


BASE Sa— SLOPE 1 to 1. 




Add. 


1 


Add. 




Dedoct 


ssas 


Deduct. 


03 




» 




a*" 
1 




a 




1 


.6111 


31 


36.1666 


.0062 


31 


6.9321 


2 


1.2592 


32 


37.9259 


2 


.0247 


32 


6.321U 


3 


1.9444 


33 


39.7222 


3 


.0565 


33 


6.7222 


4 


2.6666 


94 


41.5555 


4 


.0988 


34 


7.1358 


5 


3.4259 


35 


43.^^59 


6 


.1643 


36 


7.6617 


6 


4.2222 


36 


45.3333 


6 


.2222 


36 


8.0000 


7 


5.0555 


37 


47.2777 


7 


.3025 


37 


8.4506 


8 


5.9259 


38 


49.2592 


8 


.3951 


38 


8.9135 


9 


6.8333 


39 


51.2777 


9 


.5000 


39 


9.3888 


10 


7.7777 


40 


53.3333 


10 


.6173 


40 


9.8765 


11 


8.7592 


41 


55.4259 


11 


.7469 


41 


10.3765 


12 


9.7777 


42 


57.5655 


12 


.8889 


42 


10.8888 


13 


10.8333 


43 


69.7222 


13 


1.0432 


48 


11.4135 


14 


11.9259 


44 


61.9259 


14 


1.2099 


44 


11.9606 


16 


13X)555 


45 


64.1666 


15 


1.3889 


46 


12JS000 


16 


14.2222 


46 


66.4444 


16 


1.6802 


46 


ia0617 


17 


15.4259 


47 


68.7692 


17 


1.7839 


47 


13.6358 


18 


16.6666 


48 


71.1111 


18 


2.0000 


48 


142222 


19 


17.9444 


4l» 


73.6000 


19 


2.2284 


49 


14.8-209 


20 


19.2592 


50 


75.9259 


20 


2.4691 


50 


16.4321 


21 


20J6111 


51 


78.3888 


21 


2.7222 


51 


16.0555 


22 


22J0OOO 


52 


80.8888 


22 


2.9876 


52 


16.6913 


23 


23.4259 


53 


83.4259 


23 


3.2654 


6d 


17.3396 


24 


24.8888 


54 


86.0000 


24 


3.5556 


54 


18.0000 


25 


26.3888 


55 


88.6111 


26 


3.8580 


56 


18.6728 


26 


27.9259 


56 


91.2592 


26 


4.1728 


66 


19.3580 


27 


29.5000 


57 


93.9444 


27 


4.5000 


67 


20.0556 


28 


31.1111 


58 


96.6666 


28 


4.8395 


68 


20.7664 


29 


32.7592 


59 


99.4250 


29 


5.1913 


59 


21.4876 


30 


34.4444 60 


102.2222 


80 5.65551 


60 


22.2222 



( CXXY. ) 

BASE 32— SLOPE IJ to 1. 





Add. 




Add. 


I 


Deduct. 


Dif.of 
Hta. 


Dednct. 


1 


.6157 


31 


40.6157 


.0077 


31 


7,4151 


2 


1.2777 


32 


42.6666 


2 


.0309 


32 


7.9012 


3 


1.9861 


33 


44.7639 


3 


.0694 


33 


8.4027 


4 


2,7407 


34 


46.9074 


4 


.1234 


34 


8.9197 


5 


3.5416 


35 


^.0971 


6 


.1928 


35 


9.4521 


6 


4.3888 


36 


61.3333 


6 


.2778 


36 


10.0000 


7 


5.2823 


37 


53.6167 


7 


.3781 


37 


10.5632 


8 


6.2222 


38 


66.9444 


8 


.4938 


38 


11.1419 


9 


7.2083 


39 


58.3195 


9 


.6260 


39 


11.7361 


10 


8.2407 


40 


60.7407 


10 


.7716 


40 


12.3456 


11 


9.3195 


41 


63.2083 


11 


.9336 


41 


12.9706 


12 


10.4444 


42 


65.7222 


12 


1.1111 


42 


13.6111 


13 


11.6157 


43 


68.2823 


13 


1.3040 


43 


14.2680 


14 


12.8333 


44 


70.8888 


14 


1.5123 


44 


14.9382 


15 


14.0971 


45 


73.5416 


15 


1.7361 


46 


16.^50 


16 


16.4074 


46 


76.2407 


16 


1.9753 


46 


16.3271 


17 


16.7639 


47 


78.9861 


17 


2.2299 


47 


17.0447 


18 


18.1666 


48 


81.7777 


18 


2.5000 


48 


17.7777 


19 


19.6167 


49 


84.6167 


19 


2.7866 


49 


18.5262 


20 


21.1111 


50 


87.6000 


20 


3.0864 


60 


19.2901 


21 


22.6528 


61 


90.4305 


21 


3.4028 


61 


20.0694 


22 


24.2407 


52 


93.4074 


22 


3.7346 


62 


20.8641 


23 


25.8760 


53 


96.4305 


23 


4.0818 


63 


21.6743 


24 


27.5655 


54 


99.5000 


24 


4.4444 


54 


22.5000 


26 


29.2823 


56 


102.6157 


26 


4.8228 


65 


23.3410 


26 


31.0556 


56 


106.7777 


26 


6.2160 


56 


24.1976 


27 


32.8750 


67 


108.9861 


27 


5.^50 


67 


26.0694 


28 


34.7407 


58 


112.2407 


28 


6.0494 


58 


25.9567 


29 


36.662& 


69 


115.5416 


29 


6.4891 


69 


26.8696 


30 


38.6111 


60 


118.8888 


80 


6.9444 


60 


27.7777 



J, 



L^ 







( cxxvi. ) 




BASE 32— SLOPE IJ to 1. 


•8 


Add. 


1 


Add. 




Oednct. 




Dedact. 


S 




s 




a* 
1 




Q* 




1 


.6203 


31 


45.0648 


.0092 


31 


8.8981 


2 


1.2962 


32 


47.4074 


2 


.0370 


32 


9.4816 


3 


2.0277 


33 


49.8055 


3 


.0833 


38 


10.0838 


4 


2.8148 


34 


52.2592 


4 


.1480 


34 


10.7037 


6 


3.6574 


35 


,54.7685 


5 


.2313 


35 


11.3425 


6 


4.5555 


86 


57.3833 


6 


.3333 


36 


12.0000 


7 


5.5092 


37 


59.9536 


7 


.4537 


37 


12.6759 


8 


6.5185 


38 


62.6296 


8 


.5926 


38 


13.8708 . 


9 


7.5833 


39 


65.3611 


9 


.7600 


39 


14.0888 


10 


8.7037 


40 


68.1481 


10 


.9269 


40 


14.8148 


11 


9.8797 


41 


70.9907 


11 


1.1203 


41 


16.5648 


12 


11.1111 


42 


73.8888 


12 


1.3333 


42 


16.3833 


13 


12.3981 


43 


76.8425 


13 


1.5648 


43 


17.1202 


14 


13.7407 


44 


79.8519 


14 


1.8148 


44 


17.9259 


15 


15.1388 


45 


82.9166 


15 


2,0833 


45 


18.7600 


16 


16.5926 


46 


86.0370 


16 


2.3704 


46 


19.6925 


17 


18.1018 


47 


89.2129 


17 


2.6759 


47 


20.4537 


18 


19.6666 


48 


92.4444 


18 


3.0000 


48 


21.3333 


19 


21.2870 


49 


95.7314 


19 


3.3426 


49 


22.2314 


20 


22.9629 


50 


99.0740 


20 


3.7037 


50 


23.1481 


21 


24.6944 


61 


102.4722 


21 


4.0833 


61 


24.0833 


22 


26.4814 


52 


105.9259 


22 


4.4816 


52 


25.0370 


23 


28.3241 


63 


109.4351 


23 


4.8981 


53 


26.0092 


24 


30.2222 


54 


113.0000 


24 


6.3333 


64 


27.0000 


25 


32.1758 


55 


116.6203 


25 


5.7869 


55 


28.0092 


26 


34.1851 


56 


120.2962 


26 


6.2592 


56 


29.0370 


27 


36.2500 


57 


124.0277 


27 


6.7500 


67 


30.0833 


28 


38.3703 


58 


127.8148 


28 


7.2692 


58 


31.1481 


•29 


40.5462 


59 


131.6674 


29 


7.7869 


59 


32.2314 


30 


42.7777 


60 


135.5655 


30 


8.33a3 


60 


33.3333 





( CXZTli. ) 




BASE 32— SLOPE If to 1. 




4^ 

t 


Add. 


•r 


Add. 




Deduct. 


if. of 


Deduct. 


1 




» 




Q* 




Q" 




.6250 


31 


49.5139 


1 


.0108 


31 


10.3811 


2 


1.3147 


32 


62.1481 


2 


.0432 


32 


11.0617 


3 


2.0694 


33 


54.8472 


3 


.0972 


33 


11.7638 


4 


2.8888 


34 


57.6111 


4 


.1728 


34 


12.4876 


5 


3.7731 


36 


60.4397 


6 


i»00 


35 


13.2330 


6 


4.7222 


36 


63.3333 


6 


.3889 


36 


14.0000 


7 


5.7360 


37 


66.2916 


7 


.5293 


37 


14.7885 


8 


6.8148 


38 


69.3148 


8 


.6913 


38 


16.5987 


9 


7.9588 


39 


72.4027 


9 


.8760 


39 


16.4305 


10 


9.1666 


40 


75.5556 


10 


1.0802 


40 


17.2839 


11 


10.4399 


41 


78.7731 


11 


1.3071 


41 


18.1689 


12 


11.7777 


42 


82.0665 


12 


1.6555 


^ 


19.0655 


13 


13.1805 


43 


85.4027 


13 


1.8256 


43 


19.9747 


14 


14.6481 


44 


88.8148 


14 


2.1173 


44 


20.9136 


16 


16.1805 


45 


92.2916 


16 


2.4305 


45 


21.8750 


16 


17.7777 


46 


96.8333 


16 


2.7654 


46 


22.8580 


17 


19.4399 


47 


99.4397 


17 


3.1219 


47 


23.8626 


18 


21.1666 


48 


103.1111 


18 


3.5000 


48 


248888 


19 


22.9583 


49 


106.8472 


19 


3.8997 


49 


25.9367 


20 


24.8148 


60 


110.6481 


20 


4.8210 


60 


27.0061 


21 


26.7360 


61 


114.5139 


21 


4.7639 


61 


28.0972 


22 


28.7222 


62 


118.4444 


22 


6.2284 


52 


29.2098 


23 


30.7731 


63 


122.4399 


23 


5.7145 


53 


30.3441 


24 


32.8888 


54 


126.500J 


24 


6.2222 


64 


31.5000 


25 


36.0694 


55 


130.6250 


26 


6.7516 


65 


32.6774 


26 


37.3147 


66 


1348147 


26 


7.3024 


56 


33.8765 


27 


39.6260 


57 


139.0694 


27 


7.8750 


57 


35.0972 


28 


^0000 


58 


143.3888 


28 


8.4691 


58 


36.3394 


29 


44.4399 


69 


147.7731 


29 


9.0848 59 


37.6033 


30 


46.94441 60 

* 


152.2222 


30 


9.7222 60 


38.8888 



( cxxviii. ) 
BASE 32— SLOPE 2 to 1. 



1 


Add. 


1 


Add. 


if. of 

Ht8. 


Dedact. 




Deduct. 


ffi 




31 


• 


1 




31 




1 


.6296 


53.9629! 


,0123 


11.8642 


2 


1.3333 


32 


56.8888 


2 


.0494 


32 


12.6419 


3 


2.1111 


33 


59.8888 


3 


.1111 


33 


13.4444 


4 


2.9629 


34 


62.9629 


4 


,1975 


84 


14.2716 


5 


3.8888 


35 


66.1111 


5 


.3086 


35 


15.1234 


6 


4.8888 


36 


69.3833 


6 


.4444 


86 


16.0000 


7 


6.9629 


37 


72.6296 


7 


.6049 


37 


16.9012 


8 


7.1111 


38 


76.0000 


8 


.7901 


88 


17.8271 


9 


8.3333 


39 


79.4444 


9 


1.0000 


39 


18.7777 


10 


9.6296 


40 


82.9629 


10 


1.2346 


40 


19.7530 


11 


11.0000 


41 


86.5555 


11 


1.4936 


41 


20.7530 


12 


12.4444 


42 


90.2222 


12 


1.7778 


42 


21.7777 


13 


13.9629 


43 


93.9629 


13 


2.0864 


43 


22.8271 


14 


15.6565 


44 


97.7777 


14 


2.4197 


44 


23.9012 


16 


17.2222 


45 


101.6666 


15 


2.7778 


45 


25.0000 


16 


18.9629 


46 


105.6296 


16 


3.1605 


46 


26.1234 


17 


20.7777 


47 


109.6666 


17 


3.5679 


47 


27.2716 


18 


22.6666 


48 


113.7777 


18 


4.0000 


48 


28.4444 


19 


24.6296 


49 


117.9629 


19 


4.4568 


49 


29.6420 


20 


26.6666 


50 


122.2222 


20 


4.9882 


60 


30.8642 


21 


28.7777 


61 


126.5555 


21 


5.4444 


61 


32.1111 


22 


30.9629 


52 


130.9629 


22 


5.9753 


62 


33.3827 


23 


33.2222 


53 


135.4444 


23 


6.6309 


53 


34.6790 


24 


35.5555 


54 


140.0000 


24 


7.1111 


54 


36.0000 


26 


37.9629 


55 


144.6296 


25 


7.7160 


65 


37.3456 


26 


40.4444 


56 


149.3333 


26 


8.3467 


56 


38.7160 


27 


43.0000 


57 


154.1111 


27 


9.0000 


67 


40.1111 


28 


45.6296 


58 


158.9629 


28 


9.6790 


58 


41.5308 


29 


48.3333 


59 


163.8888 


29 


10.3827 


69 


42.9753 


30 


51.1111 


60 


168.8888 


30 


11.1111 


60 


44.4444 







( cxxix. ) 




BASE 32— SLOPE 2j^ to 1. 


•8 


Add. 


•** 

t 


Add. 


if. of 


Deduct. 


if. of 
Hts. 


Dedaet. 


S 
1 




SB 




Q 




Q 




.6388 


31 


62.8611 


1 


.0154 


31 


14.8302 


2 


1.3703 


32 


66.8703 


2 


.0617 


32 


16.8016 


3 


2.1944 


33 


69.9722 


3 


.1389 


33 


16.8055 


4 


3.1111 


34 


7^6666 


4 


.2468 


34 


17.8395 


5 


41203 


35 


77.4637 


6 


.3867 


36 


18.9043 


6 


5.2222 


36 


81.3333 


6 


.5665 


36 


20.0000 


7 


6.4166 


37 


85.3055 


7 


.7661 


87 


21.1265 


8 


7.7087 


38 


89.3703 


8 


.9876 


38 


22.2839 


9 


9.0833 


39 


93.5277 


9 


1.2600 


39 


23.4722 


10 


10.5555 


40 


97.7777 


10 


1.5432 


40 


24.6913 


11 


12.1203 


41 


102.1203 


11 


1.8673 


41 


25.9418 


12 


13.7777 


42 


106.5555 


12 


2.2222 


42 


27.2222 


13 


15.5-^77 


43 


111.0833 


13 


2.6080 


43 


28.6360 


14 


17.3703 


44 


116.7037 


14 


3.0247 


44 


29.8766 


Id 


19.3055 


45 


120.4166 


15 


3.4722 


45 


31.2600 


16 


21.3333 


46 


125.2222 


16 


3.9506 


46 


32.6648 


17 


23.4537 


47 


130.1203 


17 


4.4699 


47 


34.0895 


18 


25.6666 


48 


135.1111 


18 


5.0000 


48 


35.5666 


19 


27.9722 


49 


140.1944 


19 


5.6710 


49 


37.0624 


20 


30.3703 


60 


146.3703 


20 


6.1728 


50 


38.5802 


21 


32.8611 


51 


150.6388 


21 


6.8065 


61 


40.1388 


22 


35.4444 


62 


156.0000 


22 


7.4691 


52 


41.7283 


23 


38.1203 


53 


161.4637 


23 


8.1636 


53 


43.3487 


24 


40.8888 


54 


167.0000 


24 


8.8889 


54 


45.0000 


25 


43.7500 


55 


172.6388 


25 


9.6456 


66 


46.6820 


26 


46.7037 


56 


178.3703 


26 


10.4321 


66 


48.3960 


27 


49.7500 


57 


184.1944 


27 


11.2600 


57 


60.1388 


28 


52.8888 


58 


190.1111 


28 


12.0587 


68 


51.9135 


29 


66.1203 


59 


196.1203 


29 


12.9782 


69 


63.7191 


30 


59.4444 


60 


202.2222 


30 


13.8889 


60 

s 


65.6666 



( cxxx. ) 


BASE 32— SLOPE 3 to 1. 


1 


Add. 


1 


Add. 


if. of 
Bts. 


Dednct. 


°i 

rax 


Deduct. 


v 




BB 




Q 




a 




1 


.6480 


31 


71.7692 


1 


.0186 


31 


17.7963 


2 


1.4074 


32 


75.8618 


2 


.0740 


32 


18.9630 


3 


2.2777 


33 


80.0565 


3 


.1667 


33 


20.1666 


4 


3.2592 


34 


843703 


4 


.2963 


34 


21.4074 


6 


43518 


35 


88.7962 


6 


.4628 


35 


22.6861 


6 


5.5555 


36 


93.3333 


6 


.6667 


36 


240000 


7 


6.8703 


37 


97.9814 


7 


.9074 


37 


25.3518 


8 


8.2962 


38 


102.7407 


8 


1.1852 


38 


26.7407 


9 


9.8333 


39 


107.6111 


9 


1.5000 


39 


28.1666 


10 


11.4814 


40 


112.6926 


10 


1.8518 


40 


29.6296 


11 


13.2407 


41 


117.6851 


11 


2.2407 


41 


31.1296 


12 


15.1111 


42 


122.8888 


12 


2.6667 


42 


32.6666 


13 


17.0926 


43 


128.2037 


13 


3.1296 


43 


34.2407 


14 


19.1851 


44 


133.6296 


14 


3.6296 


44 


35.8518 


15 


21.3888 


45 


139.1666 


15 


41667 


45 


37.5000 


16 


23.7037 


46 


1448148 


16 


47407 


46 


39.1851 


17 


26.1296 


47 


150.6740 


17 


6.3518 


47 


40.9078 


18 


28.6666 


48 


156.4444 


18 


6.0000 


48 


42.6666 


19 


31.3148 


49 


162.4269 


19 


6.6862 


49 


44.4629 


20 


34.0740 


50 


168.6185 


20 


7.4074 


50 


46.2963 


21 


36.9444 


61 


1747222 


21 


8.1667 


51 


49.1666 


22 


39.9259 


62 


181.0370 


22 


8.9629 


62 


50.0740 


23 


43.0185 


63 


187.4^29 


23 


9.7962 


63 


62.0185 


24 


46.2222 


54 


194.0000 


24 


10.6667 


64 


540000 


25 


49.5370 


55 


200.6480 


25 


11.5741 


56 


66.0184 


26 


52.9629 


66 


207.4074 


26 


12.5184 


66 


58.0740 


27 


56.5000 


67 


2142777 


27 


13.6000 


57 


60.1666 


28 


58.1480 


68 


221.2592 


28 


146185 


58 


62.2962 


29 


63.9074 


59 


228.3618 


29 


15.5741 


69 


64.4629 


30 


67.7777 


60 


235.6566 


30 


16.6667 


60 


66.6666 





( cxxxi. ) 




BASE 33-8LOPE i to 1. 


4^ 

•s 


Add. 


i 


• 

Add. 


if. of 


Dednct. 


3^ 


Deduct. 


a 




X 




O 




Q 




1 


.6157 


31 


23.3935 


1 


.0016 


31 


1.4830 


2 


1.2407 


32 


24.2963 


2 


.00^ 


32 


1.6802 


3 


1.8760 


33 


26.2083 


3 


.0139 


33 


1.6806 


4 


2.5185 


34 


2ai296 


4 


.0246 


34 


1.7830 


5 


3.1713 


35 


27.0601 


5 


.0386 


35 


1.8904 


6 


3.8333 


36 


28.0000 


6 


.0655 


36 


2.0000 


7 


4.5046 


37 


28.9^0 


7 


.0766 


37 


2.1126 


8 


5.1862 


38 


29.9074 


8 


.0988 


38 


2.2284 


9 


5.8750 


39 


30.8760 


9 


.1250 


39 


2.3472 


10 


6.5740 


40 


31.8619 


10 


.1543 


40 


2.4691 


11 


7.2824 


41 


32.8379 


11 


.1867 


41 


2.6941 


12 


8.0000 


42 


33.8333 


12 


.2222 


42 


2.7222 


13 


8.7268 


43 


34.8379 


13 


.2608 


43 


2.8645 


14 


9.4629 


44 


36.8619 


14 


.3026 


44 


2.9876 


15 


10.2083 


46 


36.8760 


15 


.3472 


45 


3.1260 


16 


10.9629 


46 


37.9074 


16 


.3951 


46 


3.2654 


17 


11.7-268 


47 


38.9940 


17 


.4460 


47 


3.4089 


18 


12.5000 


48 


40.0000 


18 


.6000 


48 


3.6566 


19 


13.2824 


49 


41.0601 


19 


.6671 


49 


3.70^ 


20 


14.0740 


60 


42.1296 


20 


.6173 


60 


3.8680 


21 


14.8760 


51 


43.2083 


21 


.6806 


61 


4.0139 


22 


15.6862 


62 


44.2963 


22 


.7469 


52 


4,1728 


23 


16.5046 


63 


46.3935 


23 


.8163 


53 


4.33^ 


24 


17.3333 


64 


46.5000 


24 


.8889 


54 


45000 


25 


18.1713 


65 


47.6157 


26 


.9647 


55 


4.6682 


26 


19.0185 


66 


48.7407 


26 


1.0432 


66 


48395 


27 


19.8760 


67 


49.8760 


27 


1.1250 


67 


6.0139 


28 


20.7407 


68 


61.0186 


28 


1.2099 


58 


6.1913 


29 


21.6167 


59 


62.1713 


29 


1.2978 


59 


6.3719 


30 22.50001 


60 


53.3333 


30 


1,3889 


60 


6.6655 







( cxxxii. ) 


• 




BASE 33— SLOPE 


1 ^ to 1. 




1 


Add. 


1 


Add. 


[Si 


Deduct. 


if. of 
Bts. 


Dedact. 


• 


,« 




to 




ft 




ft 

31 






1 


.6203 


31 


27.8426 


1 


.0031 


2,9660 




2 


1.2592 


32 


29.0370 


2 


.0123 


32 


3,1605 




3 


1.9166 


33 


30.2500 


8 


.0278 


33 


3,3611 




4 


2.5926 


34 


31.4815 


4 


.0494 


34 


8.5679 




6 


3.2870 


35 


32/7814 


5 


.0772 


35 


3,7808 




6 


40000 


36 


340000 


6 


.1111 


36 


40000 




7 


4.7314 


37 


35.2870 


7 


.1512 


37 


42253 




8 


5.4815 


38 


36,5926 


8 


.1975 


38 


44568 




9 


6.2500 


39 


37.9166 


9 


.2500 


39 


4,6944 




10 


7.0370 


40 


39.2592' 

1 


10 


.3086 


40 


49383 




11 


7.8^6 


41 


40.6203 


11 


.3734 


41 


5.1883 




12 


8.6666 


42 


42.0000 


12 


AAAA 

• J. X X'X 


42 


5.4444 




13 


9.5092 


43 


43.3981 


13 


.5216 


43 


6,7076 




14 


10.3703 


44 


44.8148 


14 


.6049 


44 


5.9763 




15 


12.2500 


45 


46.2500 


15 


.6944 


45 


6.2500 




16 


12.1481 


46 


47.7037 


16 


.7901 


46 


6.5308 




17 


13.0647 


47 


49.1758 


17 


.8920 


47 


6.8179 




18 


14.0000 


48 


50.6666 


18 


1.0000 


48 


7.1111 




19 


14,9537 


49 


52.1758 


19 


1.1142 


49 


7.4105 




20 


15.9259 


50 


53.7037 


20 


1.2346 


50 


7.7160 




21 


16.9166 


51 


55.2500 


21 


1.3611 


51 


8.0277 




22 


17.9269 


52 


56.8148 


22 


1.4938 


52 


8.3456 




23 


18.9537 


53 


58.3981 


28 


1.6327 


53 


8,6697 




24 


20.0000 


54 


60.0000' 


24 


1.7778 


64 


9,0000 




25 


21.0647 


55 


61.6203 


25 


1.9295 


65 


9.3364 




26 


22.1481 


56 


63.2592 ' 


26 


2.0864 


56 


9.6790 




27 


23.2500 


57 


64.9166 


27 


2.2500 


57 


10.0277 




28 


24.3703 


58 


66.5926 


28 


2.4197 


68 


10.3827 




29 


25.50^2 


59 


68.2870 


29 


2,5956 


59 


10.7438 




30 


26.6666 


60 


70.0000' 


30 


2,7778 

• 


60 11,1111 





• 


( cxxxiii. ) 


BASE 33— SLOPE } to 1. 


4^ 

-a, 


Add. 


1 


Add. 


if. of 


Deduct. 


if. of 

Hts. 


Deduct. 


1 




X 




1 


• 


31 




.6250 


31 


32.2916 


.0046 


44490 


2 


1.2777 


32 


33.7777 


2 


.0185 


32 


4.7407 


3 


1.9583 


33 


36.2916 


3 


.0416 


33 


5.0416 


4 


2.6666 


34 


36.8333 


4 


.0740 


34 


6.3618 


5 


3.4027 


36 


38.4027 


6 


.1167 


36 


6.6712 


6 


4.1666 


36 


40.0000 


6 


.1667 


36 


6.0000 


7 


4.9583 


37 


41.6250 


7 


.2268 


37 


6.3379 


8 


6.7777 


38 


43.2777 


8 


.2963 


38 


6.6851 


9 


6.6260 


39 


44.9583 


9 


.3760 


39 


7.0416 


10 


7.6000 


40 


46.6666 


10 


.4630 


40 


7.4074 


11 


.8.4027 


41 


48.4027 


11 


.5602 


41 


7.7824 


12 


9.3333 


^ 


60.1666 


12 


.6667 


42 


8.1666 


13 


10.2916 


43 


51.9583 


13 


.7824 


43 


8.6602 


14 


11.2777 


44 


53.7777 


14 


.9074 


44 


8.9629 


16 


12.2916 


46 


55.6250 


15 


1.0417 


46 


9.3750 


16 


13.3333 


46 


67.5000 


16 


1.1852 


46 


9.7962 


17 


14.4027 


47 


59.4027 


17 


1.3379 


47 


10.2268 


18 


15.5000 


48 


61.3333 


18 


1.5000 


48 


10.6666 


19 


16.6260 


49 


63.2916 


19 


1.6713 


49 


11.1157 


20 


17.7777 


60 


66.2777 


20 


1.8518 


60 


11.6740 


21 


18.9583 


61 


67.2916 


21 


2.0417 


61 


12.0416 


22 


20.1666 


62 


69.3333 


22 


2.2407 


62 


12.5186 


23 


21.4027 


53 


71.4027 


23 


2.4491 


53 


13.0046 


24 


22.6666 


54 


73.6000 


24 


2.6667 


54 


13.5000 


26 


23.9583 


65 


76.6250 


26 


2.8936 


55 


14.0046 


26 


26.2777 


56 


77.7777 


26 


3.1296 


66 


14.5186 


27 


26.6250 


57 


79.9583 


27 


3.3750 


67 


15.0416 


28 


28.0000 


58 


82.1666 


,28 


3.6296 


68 


16.6740 


29 


29.4027 


59 


84.4027 


29 


3.8936 69 


16.1167 


30 130.8333 


60 


86.6666 


30 


4.1667 60 


16.6666 



( cxzsiv. ) 
BASE 33— SLOPE 



1 


Add. 


i 


Add. 


1 


^ 


1 


Li.<t. 


1 


.6296 


31 


36.7467 


1 


.6062 


IT 


6.9321 


2 


1.2962 


32 


38.5186 


2 


.0247 


32 


a3210 


3 


20000 


33 


40.3333 


3 


.0565 


33 


67222 


4 


2.7407 


34 


42.1861 


4 


.0988 


34 


7.1358 


5 


3.5185 


35 


440740 


5 


.1543 


35 


7.6617 


e 


43333 


36 


46.0000 


6 


.2222 


36 


8.0000 


7 


51851 


37 


47.9629 


7 


.3025 


37 


8.4506 


8 


6.OT40 


38 


40.9629 


8 


.3951 


38 


8.9135 


9 


7.0000 


39 


52.0000 


9 


.5000 


39 


9.3888 


10 


7.9629 


40 


540740 


10 


.6173 


40 


9.8765 


11 


8.9629 


41 


56.1851 


11 


.7469 


41 


10.3765 


12 


10.0000 


42 


68.3333 


12 


.8889 


42 


10.8888 


13 


11.0740 


43 


60.6185 


13 


1.0432 


43 


11.4135 


14 


12.1851 


44 


62.7407 


14 


1.2099 


44 


11.9506 


15 


13.3333 


45 


65.0000 


15 


1.3889 


46 


12.5000 


16 


145185 


46 


67.2962 


16 


1.5802 


46 


13.0617 


17 


15.7407 


47 


69.6296 


17 


1.7839 


47 


13.6358 


18 


17.0000 


48 


720000 


18 


2.0000 


48 


14.2222 


19 


18.2962 


49 


74.4074 


19 


2.2284 


49 


14.8209 


20 


19.6296 


50 


76.8518 


20 


2.4691 


50 


16.4321 


21 


21.0000 


51 


79.3333 


21 


2.7222 


51 


16.0.')55 


22 


22.4074 


52 


81.8518 


22 


2.9876 


52 


16.6913 


23 


23.8618 


63 


84.4074 


23 


3.2654 


63 


17.3395 


24 


25.3333 


54 


87.0000 


24 


3.5555 


64 


18.0000 


2S 


26.8518 


55 


89.6296 


26 


3.8530 


65 


18.6728 


26 


28.4674 


56 


92.2962 


26 


4.1728 


56 


19.3580 


27 


30.0000 


57 


96.0000 


27 


4.6000 


57 


20.0555 


28 


31.6296 


68 


97.7407 


28 


4.8395 


68 


20.7654 


29 


33.2962 


59 


100.6186 


29 


6.1913 


69 


21.4876 


30 


36.0000 


60 


103.3333 


30 


6.5556 


60 


22.2222 







( CXXXT. ) 




BASE 33— SLOPE IJ to 1. 




1' 
•a 


Add. 


1 


Add. 


".a 


Dedact. 


if. of 
Hts. 


Deduct. 




1 




s 




o 




« 






.6342 


31 


41.1898 


1 


.0077 


31 


7.4161 


] 


2 


1.3148 


32 


43.2692 


2 


.0309 


32 


7.9012 




3 


2.0417 


33 


45.3760 


3 


.0694 


33 


8.4027 


'^. 


4 


2.8148 


34 


47.6370 


4 


.1234 


34 


8.9197 


1' 

). 


5 


3.6342 


35 


49.7453 


6 


.1928 


35 


9.4521 


Oi^ 


6 


4.5000 


36 


62.0000 


6 


.2778 


36 


10.0000 


i 


7 


5.4120 


37 


54.3009 


7 


.3781 


37 


10.6632 


)» 


8 


6.3703 


38 


66.6481 


8 


.4938 


38 


11.1419 


3S 


9 


7.3760 


39 


69.0417 


9 


.6260 


39 


11.7361 





10 


8.4258 


40 


61.4814 


10 


.7716 


40 


12.3456 


Si 


11 


9.6231 


41 


63.9676 


11 


.9336 


41 


12.9706 




12 


10.6666 


42 


66.6000 


12 


1.1111 


42 


13.6111 




13 


11.8664 


43 


69.0786 


13 


1.3040 


43 


142680 


If 


14 


13.0926 


44 


71.7037 


14 


1.5123 


44 


149382 




16 


14.3760 


45 


74.3750 


15 


1.7361 


45 


15.6260 







16 


15.7037 


46 


77.0926 


16 


1.9753 


46 


16.3271 


17 


17.0786 


47 


79.8664 


17 


2.2299 


47 


17.0447 


18 


18.5000 


48 


82.6666 


18 


2.5000 


48 


17.7777 


19 


19.9676 


49 


85.6231 


19 


2.7866 


49 


18.5262 


20 


21.4814 


60 


88.4258 


20 


3.0864 


60 


19.2901 


m 

0. 



21 


23.0417 


51 


91.3750 


21 


3.4028 


51 


20.0694 


22 


24.6481 


52 


94.3703 


22 


3.7346 


62 


20.8641 


23 


26.3009 


53 


97.4120 


23 


4.0818 


53 


21.6743 


24 


28.0000 


54 


100.6000 


24 


4.4444 


64 


22.6000 


25 


29.7453 


55 


103.6342 


25 


4.8228 


56 


23.3410 


> 


26 


31.6370 


66 


106.8148 


26 


5.2160 


66 


24.1975 


27 


33.3750 


57 


110.0417 


27 


6.6260 


67 


25.0694 


28 


35.2592 


68 


113.3148 


28 


6.0494 


68 


26.9567 


29 


37.1898 


59 


116.6342 


29 


6.4891 


59 


26.8596 


30 


39.1666 


60 


120.0000 


30 


6.9444 


60 


27.7777 





( MXXVi. ) 




BASE 33-SLOPE 


IJ to 1. 


1 


Add. 


I 


Add. 


l^ 


Dedact. 


¥ 


Dedact. 


1 


.6388 


31 


45.6388 


1 


.0092 


31 


8.8981 


2 


1.3333 


32 


48.0000 


2 


.0370 


32 


9.4816 


3 


20833 


33 


.50.4166 


3 


.0833 


33 


10.0833 


4 


2,8888 


34 


62.8888 


4 


.1480 


34 


10.7037 


5 


3.7500 


36 


55.4166 


5 


.2313 


35 


11.3425 


6 


4.6666 


36 


58.0000 


6 


.33.33 


36 


120000 


7 


5.6388 


37 


60.6388 


7 


.4537 


37 


126769 


8 


6.6666 


38 


63.3333 


8 


.5926 


38 


13.3703 


9 


7.7500 


39 


06.0833 


9 


.7500 


39 


14.0833 


10 




40 


68.8888 


10 


.9259 


40 


14.8148 


11 


10.0833 


41 


71.7500 


11 


1.1203 


41 


15.5648 


12 


11.3333 


42 


74.6666 


12 


1.3333 


42 


163333 


13 


126388 


43 


77.6388 


13 


1.5648 


43 


17.1202 


14 


14.0000 


44 


806666 


14 


1.8148 


44 


17.9259 


15 


15.4166 


45 


83.7500 


15 


20833 


45 


18.7500 


16 


16.8888 


46 


86.8888 


16 


23704 


46 


19.6925 


17 


18.4166 


47 


90.0833 


17 


26759 


47 


20.4537 


18 


200000 


48 


93.3333 


18 


3.0000 


48 


21.3333 


19 


21.6388 


49 


96.6M 


19 


3.3426 


49 


22.2314 


20 


23.3333 


50 


100.0000 


20 


3.7037 


50 


23.1481 


21 


25.0833 


51 


103.4166 


21 


4.0833 


51 


24.0833 


22 


26.8888 


52 


106.8888 


22 


4.4815 


52 


25.0370 


23 


28.7500 


53 


1104166 


23 


4.8981 


63 


26.0092 


24 


30.6666 


54 


114.0000 


24 


5.3333 


64 


27.0000 


25 


326388 


55 


117.6388 


26 


5.7869 


65 


28.0092 


26 


34.6666 


56 


121.3333 


26 


6.2592 


66 


29.0370 


27 


36.7500 


67 


125.0833 


27 


6.7600 


67 


30.0833 




3-1.8888 


58 


128.8888 


28 


7.2592 


58 


31.1481 




41.0833 


59 


132.7500 


29 


7.7869 


59 


32.2314 




43.3833 


60 


136.6866 


30 


8.3333 


60 


33.3333 



( cxxxvii. ) 
BASE 33— SLOPE If to 1. 





Add. 




Add. 


if. of 
Hts. 


Deduct. 


if. of 
Hts. 


Dedu<A. 


B 




X 




P 




o 




1 


.6435 


31 


60.0879 


1 


.0108 


31 


10.3811 


2 


1.3518 


32 


52.7407 


2 


.0432 


32 


11.0617 


3 


2.1260 


33 


56.4583 


3 


.0972 


33 


11.7638 


4 


2.9629 


34 


58.2407 


4 


.1728 


34 


12.4876 


5 


3.8657 


36 


61,0879 


6 


.2700 


35 


13.2330 


6 


4.8333 


36 


64.0000 


6 


.3889 


36 


14.0000 


7 


6.8657 


37 


66.9768 


7 


.5293 


37 


14.7885 ■ 


8 


6.9629 


38 


70.0186 


8 


.6913 


38 


15.5987 1 


9 


8.1250 


99 


73.1260 


9 


.8760 


39. 


16.4305 1 


10 


9.3518 


40 


76.2962 


10 


1.0802 


40 


17.2839 


11 


10.6435 


41 


79.6324 


11 


1.3071 


41 


18.1589 


12 


12.0000 


42 


82.8333 


12 


1.6555 


42 


19.0566 


13 


13.4214 


43 


86.1991 


13 


1.8256 


43 


19.9747 


14 


14.9074 


44 


89.6296 


14 


2.1173 


44 


20.9136 


16 


16.4583 


45 


93.1250 


15 


2.4305 


45 


21.8750 


16 


18.0740 


46 


96.6851 


16 


2.7654 


46 


^.8580 


17 


19.7546 


47 


100.3101 


17 


3.1219 


47 


23.8626 


18 


21.5000 


48 


104.0000 


18 


3.5000 


48 


24.8888 


19 


23.3101 


49 


107.7546 


19 


3.8997 


49 


25.9367 


20 


25.1851 


60 


111.6740 


20 


4.3210 


60 


27.0061 


21 


27.1250 


61 


115.4683 


21 


4.7639 


61 


28.0972 


22 


29.1296 


62 


119.4074 


22 


5.2284 


62 


29.2098 


23 


31.1991 


63 


123.4214 


23 


5.7145 


63 


3a3441 


24 


33.3333 


54 


127.6000 


24 


6.2222 


64, 


31.5000 '■ 


25 


36.5324 


55 


131.6436 


26 


6.7516 


55 


32.6774 ! 


26 


37.7962 


66 


136.8518 


26 


7.3024 


56 


33.8765 


27 


40.1250 


57 


140.1260 


27 


7.8750 


57 


35.0972 


28 


42.5185 


68 


144.4629 


28 


8.4691 


68 


36.3394 


29 


44.9768 


59 


148.8667 


29 


9.0848 


59 


37.6033 


30 


47.5000 60 153.3333] 


30 


9.7222 


60 38.8888 ■■. 

T 





( cxxxviii. ) 






BASE 33-8LOPE 2 to 1. 


■ 


} 

X 


Add. 


1 


Add. 


a* 


Dedact. 


saai 
O 

31 


Deduct. 


1 


.6481 


31 


546370 


1 


.0123 


11.8642 


2 


1.3708 


32 


67.4814 


2 


.0494 


32 


12.6419 


3 


2.1666 


33 


60.6000 


3 


.1111 


33 


13.4444 


4 


3.0370 


34 


63.6926 


4 


.1976 


34 


14.2716 


5 


3.9814 


36 


66.7692 


5 


.3086 


36 


15.1234 


6 


6.0000 


36 


70.0000 


6 


.4444 


36 


16.0000 


7 


6.0926 


37 


73.3148 


7 


.6049 


37 


lft9012 


8 


7.2592 


38 


76.7037 


8 


.7901 


38 


17.8271 


9 


8.6000 


39 


80.1666 


9 


1.0000 


39 


18.7777 


10 


9.8148 


40 


83.7037 


10 


1.2346 


40 


19.7530 


H 


11.2037 


41 


87.3148 


11 


1.4938 


41 


20.7530 


12 


12.6666 


42 


91.0000 


12 


1.7778 


42 


21.7777 


13 


14.2037 


43 


94.7592 


13 


2.0864 


43 


22.8271 


14 


16.8148 


44 


98.6926 


14 


2.4197 


44 


23.9012 


16 


17.6000 


46 


102.6000 


16 


2.7778 


46 


26.0000 


16 


19.2592 


46 


106.4814 


16 


3.1605 


46 


26.1234 


17 


21.0926 


47 


110.6370 


17 


3.6679 


47 


27.2716 


18 


23.0000 


48 


114.6666 


18 


4.0000 


48 


28.4444 


19 


24.9814 


49 


118.9703 


19 


4.4668 


49 


29.6420 


20 


27.0370 


50 


123.1481 


20 


4.9382 


60 


30.86^ 


21 


29.1666 


61 


127.5000 


21 


6.4444 


61 


32.1111 


22 


31.3703 


62 


131.9269 


22 


5.9753 


62 


33.3827 


23 


33.6481 


63 


136.4259 


23 


6.5309 


63 


34.6790 


24 


36.0000 


64 


141.0000 


24 


7.1111 


64 


36.0000 


25 


38.4269 


66 


145.6481 


26 


7.7160 


65 


37.3466 


26 


40.9269 


66 


150.3703 


26 


8.3467 


56 


38.7160 


27 


43.6000 


57 


166.1666 


27 


9.0000 


67 


40.1111 


28 


46.1481 


68 


160.0370 


28 


9.6790 


58 


41.6308 


29 


48.8703 


69 


164.9814 


29 


10.3827 


69 


42.9763 


30 


61.6666 


60 


170.0000 


30 


11.1111 


60 


44.4444 







( cxxxix. ) 




BASE 33— SLOPE 2i to 1. 


■a 

'S 

s 

1 


Add. 


1 

s 


Add. 


Dif.of 
Hte. 


Deduct, 


Dif,of 
Hts, 


Dedoct 


,6674 


31 


63.4352 


1 


,0164 


31 


14,8302 


2 


1.4074 


32 


66.9629 


2 


,0617 


32 


16,8013 


3 


2.2500 


33 


70.5833 


3 


.1389 


S3 


16.8065 


4 


3.1852 


34 


74.2963 


4 


.2468 


34 


17,8396 


5 


4.2129 


35 


78.1018 


5 


,3857 


36 


18.9043 


6 


5.3333 


36 


82,0000 


6 


,5666 


36 


20,0000 


7 


&5463 


37 


86.9907 


7' 


,7561 


37 


21,1266 


8 


7.8518 


38 


90.0740 


8 


,9876 


38 


22.2839 


9 


9.2500 


39 


94.2600 


9 


1,2500 


39 


23,4722 


10 


10.7407 


40 


98,6186 


10 


1,6432 


40 


24.6913 


11 


12.3241 


41 


102.8796 


11 


1.8673 


41 


25,9413 


12 


14.0000 


42 


107,3333 


12 


2,2222 


42 


27^2222 


13 


15.7685 


43 


111.8796 


13 


2,6080 


43 


2a5360 


14 


17.6296 


44 


116.6185 


14 


3.0247 


44 


29.8765 


15 


19.5833 


46 


121,2600 


15 


3,47^ 


45 


31.2600 


16 


21.6296 


46 


126.0740 


16 


3.9606 


46 


32.6543 


17 


24.7685 


47 


130.9907 


17 


4.4599 


47 


34,0895 


18 


26.0000 


48 


136.0000 


18 


6,0000 


48 


36.6665 


19 


28.3241 


49 


141,1018 


19 


5,5710 


49 


37,0524 


20 


30.7407 


50 


146.2963 


20 


&1728 


50 


38.5802 


21 


33.2600 


51 


151,5833 


21 


6.8056 


51 


40.1388 


22 


35.8518 


62 


156.9629 


22 


7.4691 


62 


41,7283 


23 


38.5463 


53 


162,4362 


23 


8.1636 


63 


43.3487 


24 


41.3333 


54 


168.0000 


24 


8.8889 


64 


45.0000 


25 


44.2129 


65 


173.6574 


25 


9.6466 


65 


46,6820 


26 


47.1852 


56 


179.4074 


26 


10.4321 


66 


48,3960 


27 


50.2600 


67 


185.2600 


27 


11.2500 


67 


60,1388 


28 


53.4074 


58 


191.1862 


28 


12,0587 


58 


51,9135 


29 


56.6674 


59 


197,2129 


29 


12.9782 


69 


53,7191 


30 


60.0000 


60 


203.3333 


30 


13,8889 


60 


66.5656 



( cxl. ) 
BASE 33— SLOPE 3 to 1. 



» 



1 
2 
3 
4 
6 

6 
7 
8 
9 
10 

11 
12 
13 
14 
15 

16 
17 
18 
19 
20 

21 
22 
23 
24 
26 

26 
27 

28 
29 
30 



Add. 



.6666 
1.4444 
2.3333 
3.3333 
4.4444 

5.6666 

7.0000 

8.4444 

10.0000 

11.6666 

13.4444 
15.3333 
17.3333 
19.4444 
21.6666 

24.0000 
27.4444 
29.0000 
31.6666 
34.4444 

37.3333 
40.3333 
43.4444 
46.6666 
50.0000 

53.4444 
57.0000 
60.6666 
64.4444 
68.3333 






31 
32 
33 
34 
35 

36 
37 
38 
39 
40 

41 
42 
43 
44 
45 

46 

47 
48 
49 
50 

51 
52 
53 
54 
^ 

56 
57 
58 
59 
60 



Add. 



72.3333 
76.4444 
80.6666 
85.0000 
89.4444 

94,0000 

98.6666 

103.4444 

108.3333 

113.3333 

118.4444 
123.6666 
129.0000 
134.4444 
140.0000 

145.6666 
151.4444 
157.3333 
163.3333 
169.4444 

175.6666 
182.0000 
188.4444 
195.0000 
201.6666 

208.4444 
215.3333 
222.3333 
229.4444 
236.6666 



Q 



1 
2 
3 
4 
6 

6 
7 

8 

9 

10 

11 
12 
13 
14 
15 

16 
17 
18 
19 
20 

21 
22 
23 
24 
25 

26 
27 
28 
29 
30 



Deduct. 



.0185 
.0740 
.1667 
.2963 
.4628 

.6667 

.9074 

1.1852 

1.5000 

1.8518 

2.2407 
2.6667 
3.1296 
3.6296 
4.1667 

4.7407 
5.3518 
6.0000 
6.6852 
7.4074 

8.1667 

8.9629 

9.7962 

10.6667 

11.5741 

12.5184 
13.5000 
14.5185 
15.5741 
16.6667 






Deduct. 



31 
32 
33 
34 
35 

36 
37 

38 
39 
40 

41 
42 
43 
44 
45 

46 

47 
48 
49 
50 

51 
52 
53 
54 
55 

56 
57 
58 
59 
60 66.6666 



17.7963 
18.9630 
20.1666 
21.4074 
22.6851 

24.0000 
25.3518 
26.7407 
28.1666 
29.6296 

31.1296 
32.6666 
34i2407 
35.8518 
37.5000 

39.1851 
40.9074 
42.6666 
44.4629 
46.2963 

49.1666 
50.0740 
52.0185 
54.0000 
56.0184 

58.0740 
60.1666 
62.2962 
64.4629 



d 



i 



1 

1 


1 

( cxli. ) 


BASE 34— SLOPE J to 1. 


s 

K 
1 


Add. 


1 

SB 


Add. 


1 


Deduct. 


a" 

81 


Deduct. 


.6342 


31 


23.9676 


.0015 


1.4830 


2 


1.2777 


32 


24.8888 


2 


.0062 


32 


1.5802 


3 


1.9305 


33 


25.8194 


8 


.0139 


38 


1.6805 


4 


2.5926 


34 


26.7592 


4 


.0246 


84 


1.7839 


5 


3.2638 


35 


27.7083 


5 


.0386 


85 


1.8904 


6 


3.9444 


36 


28.6666 


6 


.0655 


86 


2.0000 


7 


4.63^ 


37 


29.6342 


7 


.0756 


37 


2.1126 


8 


5.3333 


38 


30.6111 


8 


.0988 


38 


2.2284 


9 


6.0416 


39 


31.5972 


9 


.1260 


89 


2.3472 


10 


6.7592 


40 


32.5926 


10 


.1643 


40 


2.4691 


11 


7.4861 


41 


83.5972 


11 


.1867 


41 


2.5941 


12 


8.2222 


42 


346111 


12 


.2222 


42 


2.7222 


13 


8.9676 


43 


36.6342 


13 


.2608 


48 


2.8645 


14 


9.7222 


44 


36.6666 


14 


.3025 


44 


2.9876 


16 


10.4861 


45 


37.7083 


15 


.3472 


46 


8.1250 


16 


11.2592 


46 


38.7592 


16 


.8951 


46 


8.2664 


17 


12.0416 


47 


39.8194 


17 


.4460 


47 


3.4089 


18 


12.8333 


48 


40.8888 


18 


.6000 


48 


3.6665 


19 


13.6342 


49 


41.9676 


19 


.6571 


49 


8.7052 


20 


14.4444 


50 


43.0565 


20 


.6178 


50 


3.8580 


21 


15.2638 


51 


44.1527 


21 


.6805 


61 


4.0139 


22 


160926 


62 


45.2592 


22 


.7469 


52 


4.1728 


23 


16.9305 


53 


46.3750 


23 


.8168 


63 


4.8349 


24 


17.7777 


64 


47.5000 


24 


.8889 


64 


4.6000 


25 


18.6342 


55 


48.6342 


26 


.9647 


55 


4.6682 


26 


19.5000 


66 


49.7777 


26 


1.0482 


56 


4.8396 


27 


2a3760 


57 


60.9305 


27 


1.1260 


57 


5.0139 


28 


21.2592 


58 


62.0926 


28 


1.2099 


58 


5.1913 


29 


22.1527 


69 


53.2688 


29 


1.2978 


59 


6.3719 


1 30 


23.0555 


60 


64.4444 


30 


1.3889 60 


6.6665 



( cxlii. ) 
BASE 34— SLOPE J to 1. 



1 


Add. 


«2 Height 


1 


.6388 


2 


1.2962 


32 


a 


1.9722 


33 


4 


2.6666 


34 


5 


3.3796 


35 


6 


41111 


36 


7 


4.8611 


37 


8 


5.6296 


38 


9 


6.4166 


39 


10 


7.2222 


40 


11 


8.0462 


41 


12 


8.8888 


42 


13 


9.7500 


43 


14 


10.6296 


44 


15 


11.6277 


45 


16 


12.4444 


46 


17 


13.3796 


47 


18 


14.3333 


48 


19 


15.3055 


49 


20 


16.2962 


50 


21 


17.3055 


51 


22 


18.3333 


52 


23 


19.3796 


53 


24 


20.4444 


54 


25 


21.5277 


55 


26 


22.6296 


56 


27 


23.7500 


57 


28 


24.8888 


58 


29 26.0462 


59 


30 


27.2222 


60 



Add. 



28.4166 
29.6296 
30.8611 
32.1111 
33.3796 

34.6666 
35.9722 
37.2962 

38.6388 
40.0000 

41.3796 
42.7777 
44.1944 
45.6296 
47.0833 

48.5555 
50.0462 
51.5555 
53.0833 
54.6296 

56.1944 

57.7777 
59.3796 
61.0000 
62.6388 

64.2962 
65.9722 
67.6666 
69.3796 
71.1111 






1 
2 
3 
4 
5 

6 
7 

8 

9 

10 

11 
12 
13 
14 
15 

16 
17 
18 
19 
20 

21 
22 
23 
24 
25 



Deduct. 



.0031 
.0123 
.0278 
.0494 
.0772 

.1111 
.1512 
.1975 
.2500 

.3734 
.4444 
.5216 
.6049 
.6944 

.7901 

.8920 

1.0000 

1.1142 

1.2346 

1.3611 
1.4938 
1.6327 
1.7778 
1.9295 



26 2.0864 



27 

28 
29 
30 



•2.2500 
2.4197 
2.5956 

2.7778 



i^ 



Deduct. 



31 
32 
33 
34 
35 

96 
37 
38 
39 
40 

41 
42 
43 
44 
45 

46 
47 
48 
49 
50 

51 
52 
53 
54 
56 

56 

57 
58 
59 
60 



2.9660 
3.1605 
3 3611 
3.5679 
3.7808 

4.0000 
4.2*253 
4.4568 
4.6944 
4.9383 

5.1883 
5.4444 
5.7067 
5.9753 
6.2500 

6.5308 
6.8179 
7.1111 
7.4105 
7.7160 

8.0277 
8.3456 
8.6697 
9.0000 
9.3364 

9.6790 
10.0277 
10.3827 
10.7438 
11.1111 







( cxliii. ) 




BASE 34 SLOPE | to 1. \ 




1 


Add. 


•4>> 

31 


Add. 


tea) 
a 

1 


Deduct. 


Dif. of 
Hts. 


1 

Deduct. 




1 


.6435 


32.8657 


.0046 


31 


4.4490 


1 


2 


1.3148 


32 


34.3703 


2 


.0185 


32 


4.7407 




3 


2.0138 


33 


36.9027 


3 


.0416 


33 


5.0416 




4 


2.7407 


34 


37.4629 


4 


.0740 


34 


6.3518 




5 


3.4953 


35 


39.0609 


6 


.1157 


35 


5.6712 




6 


4.2777 


36 


40.6666 


6 


.1667 


36 


6.0000 




7 


5.0879 


37 


42.3101 


7 


.2268 


37 


6.3379 




8 


5.9259 


38 


43.9814 


8 


.2963 


38 


6.6851 




9 


6.7916 


39 


45.6805 


9 


.3750 


39 


7.0416 




10 


7.6851 


40 


47.4074 


10 


.4630 


40 


7.4074 




11 


8.6044 


41 


49.1620 


11 


.5602 


41 


7.7824 




12 


a5565 


42 


50.9444 


12 


.6667 


42 


8.1666 




13 


10.5324 


43 


52.7546 


13 


.7824 


43 


8.5602 




14 


11.5370 


44 


64.5926 


14 


.9074 


44 


8.9629 




15 


12.5694 


45 


56.4583 


16 


1.0417 


45 


9.3750 




16 


13.6296 


46 


58.3618 


16 


1.1862 


46 


&.7962 




17 


14.7176 


47 


60.2731 


17 


1.3379 


47 


10.2268 




18 


16.8333 


48 


62.2222 


18 


1.5000 


48 


10.6666 




19 


16.9768 


49 


64.1990 


19 


1.6713 


49 


11.1157 




20 


18.1481 


60 


66.2087 


20 


1.8618 


60 


11.5740 




21 


19.5694 


51 


68.2361 


21 


2.0417 


61 


12.0416 




22 


20.5740 


52 


70.2962 


22 


2.2407 


52 


12.5185 




23 


21.8287 


53 


72.3842 


23 


2.4491 


53 


13.0046 




24 


23,1111 


54 


74.5000 


24 


2.6667 


54 


13.5000 




25 


24.4214 


65 


76.6435 


25 


28935 


65 


14.0046 




26 


25.7037 


56 


78.8148 


26 


3.1296 


56 


14.5185 




27 


27.1250 


57 


81.0138 


27 


3.3760 


57 


15.0416 




28 


28.6185 


68 


83.2407 


28 


3.6296 


58 


15.5740 




29 


29.9398 


59 


85.4953 


29 


3.8936 


59 


16.1157 




30 


31.3888 


60 


87.7777 


30 


4.1667 


60 


16.6666 





( cxliv. ) 


BASE 34^8LOPE 1 to 1. 


1 


Add. 


1 


Add. 


I 


Dednct. 




Deduct, 


1 


.6481 


31 


37.8148 


.0062 


31 


6.9321 


2 


1.3333 


32 


39.1111 


2 


.0247 


32 


a321« 


3 


2.0555 


33 


40.9444 


3 


.0555 


33 


6.7222 


4 


2.8148 


34 


42.8148 


4 


.0988 


34 


7.1358 


5 


3.6111 


35 


447222 


6 


.1543 


36 


7.5617 


6 


4.4444 


36 


46.6666 


6 


.2222 


36 


8.0000 


7 


5.3148 


37 


48.6481 


7 


.3026 


37 


8.4506 


8 


6.2222 


38 


50.6666 


8 


.395 L 


38 


8.9136 


9 


7.1666 


39 


52.7222 


9 


.5000 


39 


9.3888 


10 


8.1481 


40 


54.8148 


10 


.6173 


40 


9.8766 


11 


9.1666 


41 


56.9444 


11 


.7469 


41 


10.3765 


12 


10.2222 


42 


59.1111 


12 


.8889 


42 


10.8888 


13 


11.3148 


43 


61,3148 


13 


1.0432 


43 


11.4136 


14 


12.4444 


44 


63.6565 


14 


1.2099 


44 


11.9506 


15 


13.6111 


46 


65.8333 


16 


1.3889 


45 


12.6000 


16 


14.8148 


46 


68.1481 


16 


1.6802 


46 


13.0617 


17 


16.0556 


47 


70.5000 


17 


1.7839 


47 


13.6358 


18 


17.3333 


48 


72.8888 


18 


2.0000 


48 


14.2222 


19 


18.6481 


49 


75.3148 


19 


2.2284 


49 


14.8-209 


20 


20.0000 


50 


77.7777 


20 


2.4691 


50 


16.4321 


21 


21.3888 


61 


80.2777 


21 


2.7222 


51 


ia0555 


22 


22.8148 


52 


82.8148 


22 


2.9876 


52 


16.6913 


23 


24.2777 


63 


86.3888 


23 


3.2654 


53 


17.3395 


24 


26.7777 


54 


88.0000 


24 


3.5556 


64 


18.0000 


26 


27.3148 


65 


90.6481 


26 


3.8580 


55 


18.6728 


26 


28.8888 


56 


93.3333 


26 


4.1728 


56 


19.3580 


27 


30.6000 


57 


96.0555 


27 


4.5000 


57 


20.0555 


28 


32.1481 


58 


98.8148 


28 


4.8395 


68 


20.7664 


29 


33.8333 


59 


101.6111 


29 


6.1913 


69 


21.4876 


30 


36.5565 60 


104.4444 


30 


5.5566 


60 


22.2222 







( cxlv. ) 






BASE 34— SLOPE 1^ to 1. 


•** 

•r 

1 


Add. 


1 


Add. 


1 


Dedaot. 


s 


Deduct. 


.6527 


81 


41.7639 


.0077 


31 


7.4161 


2 


1.3518 


32 


43.8618 


2 


.0309 


32 


7.9012 


8 


2.0972 


38 


45.9861 


8 


.0694. 


38 


8.4027 


4 


2.8888 


84 


48.1666 


4 


.1234 


34 


8.9197 


5 


3.7268 


86 


60.3935 


6 


.1928 


36 


9.4621 


6 


4.6111 


86 


62.6666 


6 


.2778 


36 


10.0000 


7 


6.5416 


37 


64.9861 


7 


.3781 


87 


10.5632 


8 


&5186 


38 


57.8518 


« 


.^S38 


38 


11.1419 


9 


7.6416 


39 


59.7639 


9 


.6260 


39 


11.7861 


10 


8.6111 


40 


62.2222 


10 


.7716 


40 


12.8466 


11 


9.7266 


41 


64.7268 


11 


.9838 


41 


12.9706 


12 


ia8888 


^ 


67.2777 


12 


LllU 


42 


18.6111 


13 


12.0972 


48 


69.8760 


IS 


1.3040 


48 


14.2680 


14 


18.3618 


44 


72.5186 


14 


1.6128 


44 


14.9882 


16 


14.6627 


45 


76.2088 


16 


1.7361 


45 


16w62d0 


16 


IftOOOO 


46 


77.9444 


16 


1.9768 


46 


16.3271 


17 


17.3936 


47 


8a7268 


17 


2.2290 


47 


17.0447 


18 


18.8333 


48 


83.6656 


18 


2.5000 


48 


17.7777 


19 


20.3194 


49 


8a4306 


19 


2.7865 


49 


18.6262 


20 


21.8618 


69 


89.3618 


20 


a0864 


60 


19.2901 


21 


28.4306 


61 


92.8194 


21 


8.4028 


61 


20.0694 


22 


25.0555 


62 


95.3338 


22 


3.7846 


62 


20.8641 


28 


26.7268 


68 


98.3985 


23 


4.0818 


63 


21.6743 


24 


28.4444 


64 


101.5000 


24 


4.4444 


64 


22.6000 


25 


80.2083. 


65 


104.6627 


26 


4.8228 


66 


aa34io 


26 


32.0186 


66 


107.8618 


26 


6.2160 


66 


24.1976 


27 


3a8750 


67 


111.0972 


27 


6.6260 


67 


26.C694 


28 


36.7777 


68 


114.3888 


28 


6.0494 


68 


26.9567 


29 


87.7268 


69 


117.7268 


29 


6.4891 


50 


26^596 


30 


3a7222 


60 


12L1111 


30 


6.9444 


60 

V 


27.7777 

'i 



(oxlvi. ) 
BASE 32— SLOPE 1^ to 1. 



} 


Add. 


n 


Add. 


1 


Dednct. 


I>if.of 


Deduct. 


1 


.6574 


31 


46.2129 


.0092 


31 


8.8981 


2 


1.8708 


82 


48.6926 


2 


.0370 


32 


9.4815 


3 


2.1388 


33 


61.0277 


8 


.0633 


83 


10.0833 


4 


2.9629 


34 


53.6186 


4 


.1480 


34 


10.7087 


6 


8.8426 


35 


66.0648 


5 


.2318 


85 


11.3435 


6 


4.7777 


86 


68.6666 


6 


.3833 


86 


12.0000 


7 


5.7685 


37 


61.8240 


7 


.4537 


37 


12.6769 


8 


a8148 


38 


64.0370 


8 


.5926 


38 


18.3703 


9 


7.9166 


89 


66.8065 


9 


.7500 


89 


14.0838 


10 


9.0740 


40 


69.6296 


10 


.9259 


40 


148148 


U 


10.2870 


41 


72.6092 


11 


1.1208 


41 


15.6648 


12 


11.5555 


42 


75.4444 


12 


1.3333 


4a 


16.3338 


13 


12.8796 


48 


78.4861 


13 


1.5648 


48 


17.1202 


14 


14.2592 


44 


81.4816 


14 


1.8148 


44 


17.9259 


lis 


15.6944 


45 


84US833 


15 


2.0883 


46 


18.7600 


16 


17.1851 


46 


87.7407 


16 


2.3704 


46 


19.6926 


17 


18.7316 


47 


90.9637 


17 


2.6769 


47 


20.4637 


18 


20.3333 


48 




18 


3.0000 


48 


21.8333 


19 


21.9907 


49 


97.6463 


19 


3.3426 


^ 


22.2314 


20 


23.7037 


50 


100.9259 


20 


3.7037 


50 


23.1481 


21 


25.4722 


51 


104.8611 


21 


4.0833 


51 


24.0838 


22 


27.2963 


52 


107.8518 


22 


4.4816 


52 


26.0370 


23 


29.1760 


53 


111.3981 


23 


4.8981 


58 


26.0092 


24 


31.1111 


54 


115.0000 


24 


6.3333 


54 


27.0000 


25 


33.1018 


55 


118.6574 


25 


6.7869 


55 


28.0092 


26 


35.1481 


56 


122.8703 


26 


6.2692 


56 


29.0370 


27 


37.2500 


57 


126.1388 


27 


6.7500 


57 


30.0838 


28 


39.4074 


58 


129.9629 


28 


7.2692 


68 


31.1481 


29 


41.6203 


59 


183.8^6 


29 


7.7869 


59 


32.2314 


30 


48.8888 


60 


137,7777 


90 


8.3833 


60 


3.38388 



L 



( cxlvii. ) 


BASE 34— SLOPE 1} to 1. 


%. 




S 




■s-- 




"8.- 




•r 

a 
1 


Add. 


a 


Add. 


gS 


Dednot. 


gl 


Dednet. 


.6620 


31 


50.6620 


1 


.0106 


81 


10.3811 


2 


1.3888 


32 


53.3333 


2 


.0432 


32 


11.0617 


3 


2.1805 


33 


56.0694 


3 


.0972 


38 


11,7638 


4 


3.0370 


34 


58.8703 


4 


.1728 


34 


12.4876 


6 


3.9583 


35 


61.7361 


5 


.2700 


35 


13.2380 


6 


4.9444 


36 


64.6666 


6 


.3889 


36 


14.0000 


7 


5.9963 


37 


67.6620 


7 


.5293 


87 


14.7885 


8 


7.1111 


38 


70.7222 


8 


.6913 


38 


15.6987 


9 


8.2916 


39 


73.8472 


9 


.8760 


89 


16.4305 


10 


9.5370 


40 


77.0370 


10 


1.0802 


40 


17.2839 


11 


10.8472 


41 


80.2916 


11 


1.3071 


41 


18.1689 


12 


12.2222 


42 


83.6111 


12 


1.5665 


4& 


19.0665 


13 


13.6620 


43 


86.9953 


13 


1.8266 


43 


19.9747 : 


14 


15.1666 


44 


90.4444 


14 


2.1173 


44 


20.9135 ' 


15 


16.7361 


45 


93.9583 


15 


2.4305 


45 


21.8750 1 


16 


18.3703 


46 


97.5370 


16 


2.7654 


46 


22.a'i80 


17 


20.0694 


47 


101.1806 


17 


3.1219 


47 


23.8626 


18 


21.8333 


48 


104.8888 


18 


3.5000 


48 


24.8888 


19 


23.6620 


4Q 


108.6620 


19 


3.8997 


49 


25.9367 


20 


25.5555 


50 


112.5000 


20 


4.3210 


50 


27.0P61 


21 


27i>138 


51 


116.4027 


21 


4.7639 


51 


28.0972 


22 


29.5370 


52 


120.3703 


22 


5.2284 


52 


29.2098 


28 


31.6250 


53 


124.4027 


23 


5.7145 


53 


30.3441 


24 


33.7777 


54 


128.600J 


24 


8.2228 


54 


31.5000 


25 


35.9953 


56 


132.6620 


25 


6.7616 


55 


32.6774 


26 


38.2777 


66 


136.8888 


26 


7.3024 


56 


88.8765 


27 


40.6250 


57 


141.1805 


27 


7.8760 


57 


35.0972 


28 


43.0370 


58 


146.5870 


28 


8.4691 


58 


36.8394 


29 


45.5188 


59 


149.9583 


29 


9.0848 


59 


37.6033 


80 


48.0555 60 


154.4444 1 30 


».7222 


60 


OOaOOOO 





( czlviii. ) 




BASE 34-8LOP£ 2 to 1. 




1 

1 


Add. 


1 

n 

31 


Add. 


a" 
1 


Deduct. 




Deduct. 




•QDOO 


66.1111 


.0123 


31 


11.8642 




2 


1.4074 


32 


68.0740 


2 


.0^4 


32 


12.6419 




S 


2.2222 


88 


61.1111 


3 


.1111 


33 


18.4444 




4 


3.1111 


84 


64.2222 


4 


.1975 


84 


142716 




d 


4.0740 


36 


67.4074 


6 


.3086 


35 


16.1284 




6 


5.1111 


86 


70.6866 


6 


.4444 


36 


laoooo 




7 


6.2222 


37 


74.0000 


7 


.60^ 


37 


16.9012 




8 


7.4074 


88 


77.4074 


8 


.7901 


38 


17.8271 




9 


8.6666 


39 


80.8888 


9 


1.0000 


39 


18.7777 




10 


10.0000 


40 


844444 


10 


1.2346 


40 


I9.75S0 




11 


11.4074 


41 


88.0740 


11 


1.4938 


41 


20.7530 




12 


12.8888 


42 


91.7777 


12 


1.7778 


42 


21.7777 




13 


144444 


48 


95.5555 


13 


2.0664 


48 


22.8271 




14 


16.0740 


44 


99.4074 


14 


2.4197 


44 


23.9012 




15 


17.7777 


45 


108.8883 


15 


2.7778 


45 


25.0000 




16 


19.5556 


46 


107.3883 


16 


3.1605 


46 


S8.1234 




17 


21.4074 


47 


111.4074 


17 


8.5679 


47 


27.2716 




18 


23.8833 


48 


115.6655 


18 


40000 


48 


28.4444 




19 


25.3833 


49 


119.7777 


19 


4.4568 


49 


29.6420 




20 


27.4074 


60 


124.0740 


20 


49882 


50 


8a86^ 




21 


29.5555 


61 


129.4444 


21 


6.4444 


61 


82.1111 




22 


31.7777 


52 


132.8888 


22 


5.9753 


62 


83.3827 




23 


34.0740 


58 


137.4074 


23 


6.6309 


53 


84.67fl0 




24 


36.4444 


64 


142.0000 


24 


7.1111 


64 


36.0000 




25 


38.8888 


56 


146.6666 


$25 


7.7160 


65 


87.8456 




26 


41.4074 


66 


151.4074 


26 


8.8467 


66 


88.7160 




27 


44.0000 


67 


166i2222 


27 


9.0000 


57 


40J111 




28 


46.6666 


58 


161J111 


28 


9.6790 


68 


41i>808 




29 


49.4074 


59 


166.0740 


29 


10.8827 


69 


42.9753 




90 


52.2222160 


171.1111 


SO 


11.1111 ' 60 


44.4444 





( cxlix. ) 


BASE 34— SLOPE 2^ to 1, 


1 


Add. 


1 

a 


Add. 


Dif.of 
Hte. 

• 


DedMt 


Dif.or 
Hte. 


Dednct. 


1 


.6759 


31 


640092 


1 


J0154 


81 


148802 


2 


1.4444 


32 


67.5555 


2 


.0617 


82 


15.8015 


3 


2.3055 


33 


71.1944 


3 


.1389 


33 


16.8066 


4 


3.2592 


34 


749269 


4 


.2468 


34 


17.8896 


5 


43055 


35 


78.7500 


5 


.3857 


85 


18.9043 


6 


5.4444 


36 


82.6666 


6 


.5665 


36 


20.0000 


7 


6.6759 


37 


86.675» 


7 


.7661 


37 


21.1266 


8 


8.0000 


38 


90.7777 


8 


.9876 


38 


22.2839 


9 


9.4166 


39 


94.9722 


9 


1.2500 


39 


23.4722 


10 


10.9260 


40 


9d.2Sd2 


10 


1.5432 


40 


246913 


11 


12.6277 


41 


103.6388 


11 


1.8673 


41 


26.9413 


12 


14.2222 


42 


108.1111 


12 


2.2222 


42 


27.2222 


13 


ia0092 


43 


112.6759 


13 


2.6080 


43 


28.5860 


14 


17.8888 


44 


117.3333 


14 


3.0247 


44 


29.8765 


15 


19.8611 


45 


122.0833 


16 


3.4722 


45 


31.2600 


16 


■21.9259 


46 


126;9259 


16 


3.9506 


46 


32.6643 


17 


24.0833 


47 


131.8611 


17 


44699 


47 


34.0895 


18 


26.3333 


48 


136.8888 


18 


5.0000 


48 


36.5666 


19 


28.6759 


49 


142.0092 


19 


6.6710 


49 


37.0524 


20 


31.1111 


50 


147.2222 


20 


6.1728 


60 


88.6802 


21 


34.6388 


51 


152.5277 


21 


6.8055 


61 


40.1388 


22 


36.2592 


52 


157.9259 


22 


7.4601 


62 


41.7283 


23 


38.9722 


53 


163.4166 


23 


8.1636 


53 


43.3487 , 


24 


41.7777 


54 


169.0000 


24 


8.8889 


64 


46.0000 


25 


45.6759 


55 


174.6759 


25 


9.6455 


65 


46.6820 


26 


47.6666 


56 


180.4444 


26 


10.4321 


56 


48.3950 


27 


50.7500 


57 


186.3055 


27 


11.2500 


57 


50.1388 


28 


53.9259 


58 


192.2592 


28 


12.0587 


68 


61.9135 


29 


57.1944 


59 


198.3055 


29 


12.9782 


69 


53.7191 


30 60.5555 

1 


60 


204.4444 


30 13.8889 


60 


55.6565 





(d.) 


BASE 84— SLOPE 3 to 1. 


1 


Add. 


1 

n 


Add. 




Deduct. 




Deduct. 


1 


.6851 


31 


72.9074 


1 


.0185 


31 


17.7963 


2 


1.4814 


32 


77.0370 


2 


.0740 


32 


18.9630 


3 


2.3888 


33 


81.2777 


8 


.1667 


83 


20.1666 


4 


3.4074 


34 


85.6296 


4 


, .2963 


34 


21.4074 


6 


4.5370 


35 


90.0926 


5 


.4628 


86 


22.6851 


6 


5.7777 


36 


946666 


6 


.6667 


36 


240000 


7 


7.1296 


37 


99.3618 


7 


.9074 


87 


26.3618 


8 


8.5925 


38 


104.1481 


8 


1.1862 


38 


26.7407 


9 


10.1666 


39 


109.0555 


9 


1.5000 


39 


28.1666 


10 


11.8518 


40 


1140740 


10 


1.8518 


40 


29.6296 


11 


13.6481 


41 


119.2037 


11 


2.2407 


41 


81.1296 


12 


15.5555 


42 


124.4444 


12 


2.6667 


42 


82.6666 


13 


17.5740 


43 


129.7962 


13 


3.1296 


48 


34.2407 


14 


19.7037 


44 


136.2692 


14 


3.6296 


44 


35.8518 


16 


21.9444 


45 


140.8333 


16 


41667 


46 


37.5000 


16 


24.2962 


46 


146.6186 


16 


47407 


46 


39.1861 


17 


26.7692 


47 


152.8148 


17 


5.8618 


47 


40.9078 


18 


29.3333 


48 


158.2222 


18 


aoooo 


48 


42.6666 


. 19 


32.0185 


49 


164.2407 


19 


6.6862 


49 


444^29 


20 


34.8148 


50 


170,3708 


20 


7.4074 


50 


46.2963 


21 


37.7222 


61 


176.6111 


21 


8.1667 


51 


49.1666 


22 


40.7407 


52 


182.9629 


22 


8.9629 


52 


60,0740 


23 


43.8703 


53 


189.4259 


23 


9.7962 


63 


62.0185 


24 


47.1111 


54 


196.0000 


24 


10.6667 


64 


64.0000 


25 


50.4^9 


56 


202.6861 


26 


11.6741 


55 


6a0184 


26 


58.9259 


66 


209.4814 


26 


12.5184 


66 


58.0740 


27 


57.6000 


67 


216.3888 


27 


13.6000 


67 


60.1666 


28 


61.1861 


68 


223.4074 


28 


146185 


68 


62.2962 


29 


64.9814 


69 


230.6370 


29 


15.5741 


59 


64.4629 


30 


Do«oooo 


60 


237.7777 


30 


lft6667 


60 66.6666 





( cM. ) 








BASE 35-SLOPi 


: i to 1. 




Hdig^t 


Add. 


1 

n 


Add. 


li 

9B 


Dedoet 




Dednct. 


1 


.6527 


31 


245416 


1 


.0015 


31 


1.4880 ! 


2 


1.3148 


32 


25.4814 


2 


.0062 


32 


1.5802 


3 


1.9861 


33 


26.4305 


3 


.0189 


83 


1.6806 


4 


2.6666 


34 


27.3888 


4 


.0246 


34 


1.7889 


6 


3.3564 


35 


28.3564 


5 


.0885 


85 


1.8904 


6 


4.0555 


36 


29.3333 


6 


.0555 


36 


2.0000 


7 


4.7638 


37 


30.8194 


7 


.0756 


37 


2.1126 


8 


5.4814 


38 


31.3148 


8 


.0968 


38 


2.2284 


9 


6.2083 


39 


32.8194 


9 


.1250 


30 


2.3472 


10 


&9444 


40 


33.3383 


10 


.1548 


40 


2.4691 


11 


7.6898 


41 


34.3564 


11 


.1867 


41 


2.6041 


12 


8.4444 


4a 


35.3888 


12 


.2222 


42 


2.7222 


13 


9.2083 


43 


36.4305 


13 


.2608 


43 


2.8546 


14 


9.9814 


44 


37.4814 


14 


.3026 


44 


2.9876 


15 


10.7638 


45 


88.5416 


15 


.3472 


45 


8.1250 


16 


11.6555 


46 


89.6111 


16 


.3961 


46 


3.2664 


17 


12.3564 


47 


40.6898 


17 


.4460 


47 


8.4089 


18 


13.1666 


48 


41.7777 


18 


.6000 


48 


3.5655 


19 


13.9861 


49 


42.8750 


19 


.5571 


49 


3.7062 


20 


14.8148 


50 


48.9814 


20 


.6173 


50 


8.8680 


21 


15.6527 


51 


45.0972 


21 


.6805 


51 


4.0180 


22 


16.5000 


52 


46.2222 


22 


.7469 


52 


41728 


23 


17.3564 


53 


47.3564 


23 


.8163 


68 


48840 


24 


18.2222 


54 


48.5000 


24 


.8889 


54 


46000 : 


25 


19.0972 


55 


^.6527 


25 


.9647 


55 


46682 


26 


19.9814 


56 


50.8148 


26 


1.0432 


66 


48305 


27 


20.8750 


57 


51.9861 


27 


1.1250 


67 


6.0189 


28 


21.7777 


58 


53.1666 


28 


1.2099 


58 


6.1913 


29 


22.6898 


59 


54.3564 


29 


1.2978 


59 


6.8719 


30 


23.6111 


60 


55.5555 


30 


1.3889 


60 


5.5566 





( clii. ) 




BASE 36— SLOPE | to 1. 


1 


Add. 


1 


Add. 


Dif.of 
Htf. 


Dednot 


Dif.of 
Uts. 


Deduct 


1 


.6674 


81 


28.9907 


1 


.0031 


31 


2.9660 


2 


1.3338 


82 


30.2222 


2 


.0123 


32 


3.1605 


8 


2.0277 


88 


31.4722 


3 


.0278 


38 


3.3611 


4 


2.7407 


84 


32.7407 


4 


.0494 


84 


a6679 


6 


3.4722 


85 


34.0277 


6 


.0772 


35 


a7808 


6 


42222 


86 


36.3833 


6 


.1111 


36 


40000 


7 


49907 


87 


36.6574 


7 


.1512 


37 


42253 


8 


6.7777 


88 


38.0000 


8 


.1976 


38 


44668 


9 


&d888 


89 


39.3611 


9 


.2500 


89 


46944 


10 


7.4074 


40 


40.7407 


10 


.3086 


40 


49383 


11 


8.2500 


41 


42.1388 


11 


.3734 


41 


5.1883 


12 


9.1111 


42 


43.5666 


12 


.4444 


42 


6.4444 


13 


a9907 


43 


449907 


13 


.5216 


43 


6.7076 


14 


10.8888 


44 


.46.4444 


14 


.6049 


44 


5.9753 


15 


11.8066 


46 


47.9166 


15 


.6944 


46 


J6.2500 


16 


12.7407 


46 


4a4074 


16 


,7901 


46 


6.5306 


17 


13.6944 


47 


60.9166 


17 


.8920 


47 


6.8179 


18 


146666 


48 


52.4444 


18 


1.0000 


48 


7.1111 


19 


15.6574 


49 


53.9907 


19 


1.1143 


49 


7.4105 


20 


ia6666 


60 


66.5655 


20 


1.2346 


60 


7.7160 


21 


17.6944 


51 


67.1388 


21 


1.3611 


61 


8.0277 


22 


18.7407 


52 


58.7407 


22 


1.^38 


82 


8.3456 


23 


19.8065 


53 


60.3611 


23 


1.6327 


63 


8.6697 


24 


20.8888 


64 


62.0000 


24 


1.7778 


64 


9.0000 


25 


21.9907 


56 


63.6674 


26 


1.9296 


65 


9.3364 


26 


23.1111 


66 


65.3333 


26 


2.0664 


66 


9.6790 


27 


242500 


57 


67.0277 


27 


2.2500 


57 


10.0277 


28 


25.4074 


58 


68.7407 


28 


2.4197 


68 


10.3827 


29 


26.5888 


59 


70.4722 


29 


2.5966 


50 


10.7438 


80 


27.7777 


60 


72.2222 


30 


2.7778 60 Ill.llU 





( diii. ) 




BASE 35— SLOPE 


. f to 1. 




Add. 




Add. 


if. of 
Hts. 


Deduct. 


if. of 
Hts. 


Deduct. 


ta 




S3 




Q 




O 




1 


.6620 


31 


33.4396 


1 


.0046 


31 


4.4490 


2 


1.3518 


32 


34.8518 


2 


.0186 


32 


47407 


3 


2.0694 


33 


36.6139 


3 


.0416 


83 


5.0416 


4 


2.8148 


34 


38.0926 


4 


,0740 


34 


5.3518 


6 


3.5879 


36 


39.6990 


6 


.1167 


36 


6.6712 


6 


4.3888 


36 


41.3838 


6 


.1667 


36 


6.0000 


7 


6.2176 


37 


42.9968 


7 


.2268 


87 


6.3379 


8 


6.074IJ 


38 


44.6851 


8 


.2963 


38 


6.6861 


9 


6.9583 


89 


46.4027 


9 


.3760 


dd 


7.0416 


10 


7.8708 


40 


48.1481 


10 


.4630 


40 


7.4074 


11 


8.8102 


41 


49.9214 


11 


.5602 


41 


7.7824 


12 


9.7777 


42 


61.7222 


12 


.6667 


42 


8.1666 


. 13 


10.7731 


48 


63.6509 


13 


.7824 


48 


8.6602 


14 


11.7963 


44 


55.4074 


14 


.0074 


44 


8.9629 


16 


12.8472 


45 


67.2916 


15 


1.0417 


45 


9.3760 


16 


13.9269 


46 


59.2037 


16 


1.1852 


46 


9.7962 


17 


16.0324 


47 


61.1435 


17 


1.3379 


47 


10.2268 


18 


16.1666 


48 


63.1111 


18 


1.5000 


48 


10.6666 


19 


17.3287 


49 


65.1J066 


19 


1.6713 


49 


11.1167 


ao 


18.5186 


50 


67.1296 


20 


1.8618 


50 


11.6740 


21 


19.7361 


61 


69.1805 


21 


2.0417 


51 


12.0416 


22 


20.9816 


62 


71.2892 


22 


2.2407 


52 


12.6185 


28 


22.2546 


53 


7a3657 


23 


2.4491 


58 


18.0046 


24 


23.5655 


54 


75.5000 


24 


2.6667 


64 


13.6000 


25 


248842 


55 


77.6620 


25 


2.8935 


66 


140046 


26 


26.2407 


56 


79.8618 


26 


3.1296 


66 


14.6185 


27 


27.6260 


67 


82.0694 


27 


3.8750 


67 


16.0416 


28 


29.0370 


68 


84.3148 


28 


3.6296 


68 


16.5740 


29 


3a4768 


59 


86.6879 


29 


8.8936 


59 


1&1157 


30 

• 


31.9444 


60 


88.8888 


30 


41667 


m 

X 


16.6666 



I 


( cliv. ) 


BASE 35 SLOPE 1 to 1. j| 


1 


Add. 


1 


Add. 


if. of 
Btf. 


Dednet. 


•8^ 

«JM Deduct. 


n 

1 




m 




1 




Q 




.6666 


31 


37.8888 


.0062 


31 


6.9321 


2 


1.3703 


32 


39.7037 


2 


.0247 


32 


6.3210 


3 


2.1111 


33 


41.5556 


8 


.0555 


33 


6.7222 


4 


2.8888 


34 


43.4444 


4 


.0988 


34 


7.1368 


5 


3.7037 


85 


46.3708 


5 


.1543 


85 


7.6617 


6 


4.5555 


36 


47.3833 


6 




86 


8.0000 


7 


6.4444 


87 


49.3338 


7 


.8025 


87 


8.4506 


8 


6.3703 


38 


61.8703 


8 


.8951 


88 


8.9185 : 


9 


7.3333 


39 


63.4444 


9 


.6000 


89 


9.3888 


10 


8.3333 


40 


55.6555 


10 


.6173 


40 


9.8766 


11 


9.3703 


41 


57.7037 


11 


.7469 


41 


10.3766 


12 


10.4444 


42 


59.8888 


12 


.8889 


4a, 


10.8888 


13 


11.6655 


43 


62.1111 


13 


1.0432 


43 


11.4135 


14 


12.7087 


44 


64.8703 


14 


1.2099 


44 


11.9506 


16 


13.8888 


46 


«6.6666 


15 


1.3889 


45 


12.6000 


16 


16.1111 


46 


69.0000 


16 


1.5802 


46 


13.0617 


17 


16.3703 


47 


71.8708 


17 


1.7839 


47 


18.6358 


18 


17.6666 


48 


73.7777 


18 


2.0000 


48 


14.2222 


19 


19.0000 


49 


76.2222 


19 


2.2284 


49 


14.8209 


20 


20.3703 


60 


78.7087 


20 


2.4691 


60 


16.4321 


21 


21.7777 


51 


81.2222 


21 


2.7222 


51 


16.a%6 


22 


23.2222 


62 


83.7777 


22 


2.9876 


52 


16.6913 


23 


247037 


63 


86.8708 


23 


3.2654 


53 


17.8395 


24 


26.2222 


54 


89.0000 


24 


8.5556 


54 


18.0000 


25 


27.7777 


66 


91.6666 


26 


3.8680 


55 


18.6728 


26 


29.3703 


66 


94.3708 


26 


4.175» 


56 


19.3680 


27 


31.0000 


67 


97.1111 


27 


4.5000 


57 


20.0555 


28 


32.6666 


68 


99.8888 


28 


4.8395 


58 


20.7664 


29 


34.3703 


59 


102.7037 


29 


5.1913 


■59 


21.4876 


SO 


36.1111 


60 


105.5565 


30 


6.5555 


60 


22.2^2 



» * 




( cl^. ) 






BASE 35— SLOPE 


IJ to 1. 




1 


Add. 


S 


Add. 


If. of 


Deduct. 


3^ 


1* 

Deduct. 


# 


» 




s 
31 




Q 




Q 






1 


.6712 


42.3379 


1 


.0077 


31 


7.4161 


: 


2 


1.3888 


%2 


444444 


2 


.0309 


32 


7.9012 




3 


2.1527 


33 


4&d972 


3 


.0694 


33 


8.4027 




4 


2.9^9 


34 


4a79e2 


4 


.1-234 


34 


8.9197 




5 


a8194 


35 


51.0416 


5 


.1928 


35 


9.4S21 


> 


6 


472^ 


36 


53.3333 


6 


.2778 


36 


10.0000 


• 


7 


5.6712 


37 


55.6712 


7 


.3781 


37 


10.5632 




8 


6.6666 


38 


58.0555 


8 


.4938 


38 


11.1419 




9 


7.7083 


39 


60.4861 


9 


.6260 


39 


11.7361 




10 


8.7962 


40 


^9629 


10 


.7716 


40 


12.3466 




11 


9.9305 


41 


65.4861 


11 


.9336 


41 


12.9706 




12 


11.1111 


42 


68.0555 


12 


1.1111 


4a 


ia6111 




13 


12.3379 


43 


70.6712 


13 


1.3040 


43 


14268C 




14 


13.6111 


44 


73.3333 


14 


1.5123 


44 


149382 




IS 


14.9305 


45 


7&0416 


15 


1.7361 


45 


15.&250 




16 


1&2962 


46 


78.7962 


16 


1.9753 


46 


16.»271 




17 


17.7083 


47 


81.99I7'2 


17 


2.2299 


47 


17.0447 




18 


19.1666 


48 


84.4444 


18 


2.5000 


48 


17.7777 




19 


20.6712 


49 


87.3379 


19 


2.7856 


49 


18.5262 




20 


22.2222 


50 


90.2777 


20 


3.0864 


60 


19.2901 




21 


23.8194 


51 


93.2638 


21 


3.4028 


51 


20.0694 




22 


25.4629 


52 


96.2962 


22 


3.7346 


62 


20.8641 




23 


27.1527 


53 


99.3760 


23 


4.0818 


53 


21.6743 


' 


^ 


28.8888 


54 


102.5000 


24 


4.4444 


54 


22.5000 




26 


30.6712 


55 


105.6712 


25 


48228 


66 


23.3410 


1 


26 


32.5000 


66 


108.8888 


26 


6.2160 


66 


24.1975 




27 


343760 


57 


112.1627 


27 


6.6260 


57 


26.0694 




28 


36.2962 


58 


116.4629 


28 


6.0494 


58 


26.9667 




29 


88.2638 


59 


118.8194 


29 


6.4891 


69 


26.8595 


1 
1 


30 


40.2777 


60 


122.2222 


30 


6.9444 


60 


27.7777 







( dvi. ) 




BASE 85— SLOPE IJ to 1. 


1 


Add. 


1 


Add. 


3S 


Deduct.' 


3^ 


Dednct. 


a 




a 




o 




Q 




1 


.67d9 


31 


46.2870 


1 


.0092 


31 


8.8961 


2 


1.4074 


32 


49.1851 


2 


.0370 


% 


9.4816 


3 


2.1944 


33 


61.6888- 


a 


.0633 


33 


10.0833 


4 


3.0S70 


34 


541481 


4 


.1480 


84 


10.7037 


5 


3.9351 


35 


66,7129 


6 


.2313 


35 


11.3425 


6 


4.8888 


36 


59.3333 


6 


.3333 


86 


12.0000 


7 


5.8981 


37 


eojooaa 


7 


.4537 


37 


12.6769 


8 


6.9629 


88 


647407 


8 


.6026 


88 


ia3708 


9 


8.0833 


39 


67.5277 


9 


.7500 


39 


140838 


10 


9.2592 


40 


70.3703 


10 


.9269 


40 


148148 


11 


10.4007 


41 


7a2685 


11 


1.1203 


41 


16.6648 


12 


11.7777 


42 


76.2222 


12 


1.3333 


42 


16.3333 


13 


13.1203 


43 


79.2314 


13 


1.5648 


43 


17.1202 


14 


14.5185 


44 


82.2962 


14 


1.8146 


44 


17.9269 


15 


15.9722 


45 


85.4166 


16 


2.0833 


46 


18.7500 


16 


17.4814 


46 


8a6925 


16 


2.3704 


46 


19.6925 


17 


19.0462 


47 


91.8240 


17 


2.6759 


47 


20.4537 


18 


20.6666 


48 


95.1111 


18 


aoooo 


48 


21.3333 


19 


22.3425 


49 


98.4637 


19 


3.3426 


49 


22.2314 


20 


24.0740 


50 


101.8618 


20 


3.7087 


60 


23.1481 


21 


25.8611 


51 


106.3055 


21 


40833 


51 


24.0633 


22 


27.7087 


52 


108.8148 


22 


44816 


62 


25.0370 


23 


29.6018 


53 


112.3796 


23 


48981 


53 


26.0092 


24 


31.5555 


54 


116.0000 


24 


6.3333 


54 


27.0000, 


26 


33.5648 


55 


119.6759 


25 


5.7869 


66 


28.0092 


26 


35.6296 


56 


123.4074 


26 


6.2592 


66 


29.0370 


27 


37.7500 


67 


127.1944 


27 


6.7500 


67 


30.0833 


28 


39.9259 


68 


131.0370 


28 


7.2592 


68 


31.1481 


29 


42.1574 


69 


134.9361 


29 


7.7669 


89 


a2.2314 


30 

< 


44.4444 


60 


138.8688 


30 


a33&3 


60 


33.3333 



( chii. ) 




BASE 35— SLOPE If to 1. 


t 

s 


Add. 


Height 


Add. 


Dif. of 
Uti. 


Deduct. 


Dif. of 
Uts. 


Deduct. 


1 


.6805 


31 


51.2361 


1 


.0108 


31 


10.3811 


2 


1.4259 


32 


53.9250 


2 


.0432 


32 


11.0617 


3 


2.2361 


33 


56.6805 


3 


.0972 


33 


11.7638 


4 


3.1111 


34 


59.5000 


4 


.1728 


34 


12.4876 


6 


4.0609 


35 


62.3842 


6 


.2700 


35 


13.2330 


6 


5.0555 


36 


65.3333 


6 


.3889 


36 


14.0000 


7 


6.1250 


37 


68.3472 


7 


.5293 


37 


147885 


8 


7.2592 


3^ 


.71.^59 


8 


.6913 


38 


15.5987 


9 


8.4583 


39 


745694 


9 


.8750 


39 


l&4a06 


10 


9.7222 


40 


77.7777 


10 


1.0802 


40 


17.2839 


11 


11.0509 


41 


81.0609 


11 


1.3071 


41 


1&1689 


12 


12.4444 


42 


. 843888 


12 


1.5555 


^ 


19.0555 


13 


13.9028 


43 


87.7916 


13 


1.8256 


43 


19.9747 


14 


15.4259 


44 


91.2592 


14 


2.1173 


44 


20.9135 


15 


17.0138 


45 


947916 


15 


2.4305 


45 


21.8750 


16 


18.6666 


46 


«IC)«eJv$Oo 


16 


2.7654 


46 


22.8580 


17 


20.0509 


47 


102.0509 


17 


3.1219 


47 


23.8626 


18 


22.1666 


48 


105.7777 


18 


3.5000 


48 


248888 


19 


240138 


49 


109.5694 


19 


3.8997 


49 


25.9367 


20 


25.9259 


50 


113.4259 


20 


43210 


50 


27.0061 


21 


27.9028 


51 


117.3472 


21 


47639 


51 


28.0972 


22 


29.9444 


52 


121.3333 


22 


5.2284 


52 


29.20% 


23 


32.0509 


53 


125.3842 


23 


5.7145 


53 


30.3441 


24 


34.2222 


54 


129.5000 


24 


6.2222 


54 


31.5000 


25 


36.4583 


55 


133.6805 


25 


&7516 


55 


32.6774 


26 


38.7592 


56 


137.9259 


26 


7.3024 


56 


33.8765 


27 


41.1250 


57 


14J.2361 


27 


7.8750 


57 


35.0972 


28 


4a5dd5 


58 


146.6111 


28 


8.4691 


58 


36.3394 


29 


46.0509 


59 


151.0509 


29 


9.0848 50 


37.6033 


30 


48.6111 


60 


155.5555 


30 


9.72^ 60 


38.8888 





( clviii. ) 




BASE 35-SLOPE 2 to 1. 


1 


Add. 


1 


Add. 


s 
1 


Dedact, 


a 

31 


Deduct. 


1 


.6851 


81 


666861 


.0123 


11.8642 


2 


1.4444 


32 


68.6666 


2 


.0494 


32 


12.6419 


8 


2.2777 


33 


61.7222 


3 


.1111 


33 


18.4444 


4 


3.1851 


34 


64.8518 


4 


.1976 


34 


14.2716 


5 


4.1666 


36 


68.0656 


6 


.3086 


36 


15.1284 


6 


6.2222 


36 


71.3333 


6 


.4444 


36 


l&OOOO 


7 


&3518 


87 


74.6851 


7 


.6049 


37 


1&9012 


8 


7.5555 


38 


78.1111 


8 


.7901 


88 


17.8271 


9 


8.8333 


39 


81.6111 


9 


1.0000 


39 


18.7777 


10 


10.1851 


40 


86.1851 


10 


1.2346 


40 


19.7530 


11 


11.6111 


41 


88.8333 


11 


1.4938 


41 


20.7630 


12 


13.1111 


42 


92.5555 


12 


1.7778 


42 


21.7777 


13 


14.6861 


43 


96.3518 


13 


2.0864 


43 


22.8271 


14 


16.8333 


44 


100.2222 


14 


2.4197 


44 


23.9012 


15 


18.0555 


46 


104.1666 


15 


2.7778 


46 


25.0000 


16 


19.8518 


46 


108.1851 


16 


3.1605 


46 


26.1234 


17 


21.7222 


47 


112.2777 


17 


8.6679 


47 


27.2716 


18 


23.6666 


48 


116.4444 


18 


4.0000 


48 


28.4444 


19 


25.6851 


49 


120.6851 


19 


4.4668 


49 


29.6420 


20 


27.7777 


60 


125.0000 


20 


4.9382 


50 


30.86^ 


21 


29.9444 


51 


129.3888 


21 


5.4444 


61 


32.1111 


22 


32.1851 


52 


133.8618 


22 


6.9753 


62 


33.3827 


23 


34.5000 


53 


138.3888 


23 


6.6309 


63 


34.6790 


24 


3&8888 


64 


143.0000 


24 


7.1111 


64 


36.0000 


25 


38.2407 


56 


147.6851 


25 


7.7160 


55 


37.8456 


26 


41.8888 


56 


152.4444 


26 


8.3467 


56 


38.7100 


27 


44.5000 


57 


157.3888 


27 


9.0000 


67 


40.1111 


28 


47.1851 


68 


162.1851 


28 


9.6790 


58 


41.5308 


29 


49.9444 


59 


167.1666 


29 


10.3827 


69 


429758 


30 


52.7777 


60 


172.2222 


30 


11.1111 


60 


44.4444 



"■ 




1 


( clix. ) 




BASE 35 SLOPE 2J to 1. 


■ 


1 

93 


Add. 


1 

31 


Add. 


Dif.of 
Hts. 


Deduct. 


Dif.of 
Hts. 


Deduct. 


■ 


1 


.6944 


645833 


1 


.0164 


31 


148302 




2 


1.4814 


32 


68.1481 


2 


.0617 


32 


16.8013 


k 


3 


2.3611 


33 


71.8055 


3 


.1389 


33 


16.8056 




4 


3.3333 


34 


75.6565 


4 


.2468 


34 


17.8305 




6 


4.3981 


35 


79.3981 


5 


.3867 


35 


18.9043 \ 




6 


5.5555 


36 


83.3333 


6 


.5565 


36 


20.0000 




7 


a8056 


37 


87.3611 


7 


.7661 


37 


21.1265 




8 


8.1481 


38 


91.4814 


8 


.9876 


38 


22.2839 




9 


9.5833 


39 


96.6944 


9 


1.2500 


39 


23.4722 




10 


11.1111 


40 


100.0000 


10 


1.6432 


40 


24.6913 




11 


12.7316 


41 


104.3981 


11 


1.8673 


41 


26.9413 




12 


14.4444 


42 


108.8888 


12 


2.2222 


42 


27.2222 


i 


13 


16.2600 


43 


113.4722 


13 


2.6080 


43 


28.5360 


t 


14 


18.1481 


44 


118.1481 


14 


3.0247 


44 


29.8766 


r 


16 


20.1388 


45 


122.9166 


15 


3.4722 


46 


31.2500 




16 


22.2222 


46 


127.7777 


16 


3.9606 


46 


32.6543 




17 


24.3981 


47 


132.7316 


17 


44699 


47 


340896 


1 


18 


2&6666 


48 


137.7777 


18 


6.0000 


48 


35.5555 




19 


29.0277 


49 


1^.9166 


19 


6.5710 


49 


37.0524 


1 


20 


31.4814 


50 


148.1481 


20 


6.1728 


50 


38.6802 




21 


340277 


51 


153.4722 


21 


6.8066 


61 


40.1388 


* 


22 


3&6666 


62 


168.8888 


22 


7.4691 


62 


41.7283 




23 


39.3981 


63 


1643981 


23 


8.1636 


63 


43.3487 




24 


42.2222 


64 


170.0000 


24 


8.8889 


54 


46.0000 




25 


45.1388 


56 


176.6944 


26 


9.6465 


66 


4&6820 




26 


48.1481 


56 


181.4814 


26 


10.4321 


66 


48.3950 


1 


27 


61.2600 


57 


187.3611 


27 


11.2600 


57 


60.1388 




28 


54.4444 


58 


193.3333 


28 


12.0587 


58 


61.9135 




29 


57.7316 


69 


199.3981 


29 


12.9782 


59 


53.7191 




30 

« 


61.1111 


60 


205.5566 


30 


13.8889 


60 


66.6556 





( die. ) 


BASE 85— SLOPE 3 to 1. 


1 

n 
1 


Add. 


1 

n 


Add. 


1 


Dednot. 


til 


Deduct 


.7087 


31 


73.4814 


.0185 


81 


17.7963 


2 


1.5186 


32 


77.6296 


2 


.0740 


82 


18.9630 


3 


2.4444 


83 


81.8888 


8 


.1667 


33 


20.1666 


4 


3.4814 


34 


86.2592 


4 


.2963 


34 


21.4074 


6 


4.6296 


85 


90.7407 


5 


.4628 


35 


22.6851 


6 


5.8888 


86 


95.3833 


6 


.6667 


36 


24.0000 


7 


7.2602 


87 


100.0370 


7 


.9074 


87 


25.3518 


8 


8.7407 


38 


104.8518 


8 


1.1852 


38 


2a7407 


9 


10.3833 


39 


109.7777 


9 


1.5000 


39 


28.1666 


10 


12.0370 


40 


1148148 


10 


1.8518 


40 


29.6296 


11 


18.8518 


41 


119.9629 


11 


2.2407 


41 


31.1296 


12 


16.7777 


42 


126.2222 


12 


2.6667 


42 


32.6666 


13 


17.8148 


43 


130.6925 


13 


8.1296 


43 


34.2407 


14 


19.9629 


44 


136.0740 


14 


3.6296 


44 


35.8518 


15 


22.2222 


45 


141.6666 


15 


41667 


45 


37.6000 


16 


245925 


46 


147.3703 


16 


47407 


46 


39.1851 


17 


27.0740 


47 


153.1851 


17 


5.3518 


47 


40.9074 


18 


29.6666 


48 


169.1111 


18 


6.0000 


48 


^.6666 


19 


32.3703 


49 


165.1481 


19 


a6852 


49 


44.4629 


20 


35.1851 


50 


171.2962 


20 


7.4074 


50 


46.2963 


21 


38.1111 


61 


177.5555 


21 


8.1667 


51 


49.1666 


22 


41.1481 


62 


183.9259 


22 


8.9629 


52 


60.0740 


23 


44.2962 


53 


190.4074 


23 


9.7962 


53 


52.0185 


24 


47.5555 


54 


197.0000 


24 


10.6667 


54 


64.0000 


25 


60.9269 


55 


203.7037 


25 


11.6741 


55 


56.0184 


26 


54.4074 


66 


210.6185 


26 


12.5184 


56 


58.0740 


27 


58.0000 


67 


217.4444 


27 


13.5000 


57 


60.1666 


28 


61.7087 


68 


224.4814 


28 


145185 


68 


62.2962 


29 


65.5185 


59 


231.6296 


29 


15.6741 


59 


64.4629 


30 

1 


69.4444 


60 


238.8888 


30 


16.6667 


60 


66.6666 







( clxi. ) 




BASE 36— SLOPE i to 1. |^ 




Add. 


■a 

•s 


Add. 


•9» 


Deduct. 


if of 
Hts. 


Dednct. 


s 




s 




1 




o 
31 




1 


.6712 


31 


25.1157 


.0015 


1.4830 


2 


1.3518 


32 


2ft 0740 


2 


.0062 


32 


1.5802 


8 


2.0416 


33 


27.0416 


3 


.0139 


33 


1.6805 


4 


2.7407 


34 


28.0185 


4 


.0246 


34 


1.7839 


5 


3.44U0 


35 


29.0046 


5 


.0385 


35 


1.8904 


6 


4.1666 


36 


30.0000 


6 


.0555 


36 


2.0000 


7 


4.8935 


37 


31.<X>46 


7 


.0766 


37 


2.1126 


8 


5.6296 


38 


32.0185 


8 


.0988 


38 


2.2284 


9 


6.3750 


39 


33.0416 


9 


.1250 


39 


2.3472 


10 


7.1290 


40 


34.0740 


10 


.1643 


40 


2.4691 


11 


7.8935 


41 


35.1157 


11 


.1867 


41 


2.5941 


12 


8.666.) 


42 


36.1666 


12 


.2222 


42 


2.7222 


13 


9.4490 


43 


37.2268 


13 


.2608 


43 


2.8546 


14 


10.2407 


44 


38.2962 


14 


.30-25 


44 


2.9876 


15 


11.0416 


45 


39.3760 


15 


.3472 


46 


3.1250 


16 


11.8518 


46 


40.4629 


16 


.3951 


46 


3.2654 


17 


12.6712 


47 


41.5601 


17 


.4460 


47 


3.4089 


18 


13.5000 


48 


42.6666 


18 


.5000 


48 


3.5555 


19 


14.3379 


49 


43.7824 


19 


.5571 


49 


3.7052 


20 


15.1851 


50 


44.9074 


20 


.6173 


50 


3.8680 


21 


ia0416 


51 


46.0416 


21 


.6805 


51 


4.0139 


22 


16.9074 


o2 


47.1851 


22 


.7469 


52 


4.1728 


23 


17.7824 


53 


48.3379 


23 


.8163 


53 


4.3349 


24 


18.6666 


54 


49.6000 


24 


.8889 


54 


4.5000 


25 


19.5601 


55 


60.6712 


25 


.9647 


55 


4.6682 


26 


20.4629 


56 


51.8518 


26 


1.0432 


56 


4.8396 


27 


21.3750 


57 


53.0416 


27 


1.1260 


57 


6.0139 


28 


22.2962 


58 


642407 


28 


1.2099 


68 


6.1913 


29 


23.2268 


59 


66.449a 


29 


1.2978 


69 


5.3719 


30 


24.1666 


60 


56.6666 


30 


1.3889 


60 


5.6565 



( dxii. ) 


BASE 36— SLOPE ^ to 1. 


M Height 

1 


Add. 


Hdght 


Add. 


Dif. of 
Htf. 

1 


Dednet 


Dif. of 
Hts. 


Deduct. 


.6750 


31 


29.5648 


1 


.0031 


31 


2.9660 


3 


1.8703 


32 


30.8148 


2 


.0123 


32 


3.1606 


3 


2.0683 


38 


32.0633 


3 


.0278 


33 


33611 


4 


2.8148 


84 


33.8703 


4 


.0494 


84 


3.5679 


6 


3.5648 


36 


84.6759 


5 


.0772 


36 


3.7808 


6 


43338 


86 


36.0000 


6 


.1111 


36 


4.0000 


7 


6.1203 


87 


37.3425 


7 


.1612 


37 


4.2263 


8 


5.9250 


88 


38.7087 


8 


.1976 


38 


4.4568 


9 


6.7600 


39 


40.0838 


9 


.2500 


39 


4.6944 


10 


7.5926 


40 


41.4814 


10 


•«K/oD 


40 


4.9388 


11 


8.4687 


41 


^8981 


11 


.3734 


41 


6.1883 


12 


9.3338 


42 


44.3338 


12 


.4444 


42 


6.4444 


13 


10.2816 


48 


45.7870 


13 


.6216 


43 


6.7067 


14 


11.1481 


44 


47.2692 


14 


.6049 


44 


5.9758 


15 


12.0883 


45 


48.7500 


15 


.6944 


46 


6.2500 


16 


18.0970 


46 


50.2592 


16 


.7901 


46 


6.5308 


17 


14.0092 


47 


51.7870 


17 


.8920 


47 


ft8179 


18 


16.0000 


48 


63.3333 


18 


1.0000 


48 


7.1111 


19 


16.0092 


49 


64.8981 


19 


1.11^ 


49 


7.4105 


90 


17.0370 


50 


5&4814 


20 


1.2346 


50 


7.7160 


21 


18.0838 


51 


68.0633 


21 


1.8611 


61 


8.0277 


' 22 


19.1481 


62 


89.7087 


22 


1.4938 


62 


8.3456 


28 


20.2816 


53 


61.3425 


23 


1.6327 


53 


8.6697 


24 


21.3333 


54 


63.0000 


24 


1.7778 


54 


9.0000 


25 


22.4587 


55 


64.6769 


25 


1.9296 


66 


9.3364 


26 


28.5926 


56 


66.8703 


26 


2.0864 


66 


9.6790 


37 


24.7500 


57 


68.0838 


27 


2.2600 


67 


10.0277 


28 


26.9269 


58 


69.8148 


28 


2.4197 


68 


10.3827 


29 


27.1203 


59 


71.6648 


29 


2.5966 


69 


10.7438 


30 


28.3888 


60 


78.3333 


30 


2.7778 


60 


11.1111 





( clxiii 


•) 




BASE 36— SLOPE } to 1. 




1 
•s 


Add. 




Add. 




Deduct. 




Dednet. 




a 
1 




a 




Q 




Q 






.6805 


31 


34.0138 


1 


.0046 


31 


44^0 




2 


1.3888 


32 


a5.6556 


2 


.0185 


32 


47407 




3 


2.1250 


33 


37.1260 


3 


.0416 


33 


6.0416 




4 


2.8888 


34 


38.7222 


4 


.0740 


34 


6.3518 




5 


3.6805 


35 


40.3472 


6 


.1167 


86 


6.6712 




6 


45000 


36 


'^.0000 


6 


.1667 


36 


6.0000 




7 


6.3472 


37 


43.6806 


7 


.2268 


87 


&3379 




8 


6.2222 


38 


46.3888 


8 


.2963 


38 


6.6861 




9 


7.1250 


39 


47.1260 


9 


.3760 


39 


7.0416 




10 


8.0556 


40 


48.8888 


10 


.4630 


40 


7.4074 




11 


9.0138 


41 


60.6806 


11 


.6602 


41 


7.7824 




12 


10.0000 


42 


52.6000 


12 


.6667 


42 


8.1666 




13 


11.0138 


43 


54.3472 


13 


.7824 


43 


8.5602 




14 


12.0565 


44 


56.2222 


14 


.9074 


44 


8.9^29 




15 


13.1250 


45 


58.1250 


16 


1.0417 


45 


9^760 


■ 


16 


14.2222 


46 


60.0666 


16 


1.1852 


46 


9.7962 




17 


16.3472 


47 


62.0138 


17 


1.3379 


47 


10.2268 




18 


16.5000 


48 


64.0000 


18 


1.6000 


48 


10.6666 




19 


17.6805 


^ 


66.0138 


19 


1.6713 


49 


11.1157 




20 


18.8888 


60 


68.0555 


20 


1.8518 


60 


11.5740 




21 


20.1250 


61 


70.1250 


21 


2.0417 


61 


12.0416 




22 


21.3888 


52 


72.2222 


22 


2.2407 


52 


12.6185 




23 


22.6805 


63 


743472 


23 


2.4491 


53 


13.0046 




24 


24.0000 


64 


76.5000 


24 


2.6667 


54 


13.5000 




26 


25.3472 


55 


78.6805 


25 


28935 


65 


14.0046 




26 


26.7222 


56 


80.8888 


26 


3.1296 


66 


145186 




27 


28.1250 


67 


83.1250 


27 


3.3760 


67 


16.0416 




28 


29.5666 


58 


86.3888 


28 


3.6296 


68 


15.6740 




29 


31.0138 


69 


87.6805 


29 


3.8935 


m 


iail67 




30 


32.6000 1 60 


90.0000 


30 


41667 1 60 


1&6666 


■ 



( cxliv. ) 


BASE 36— SLOPE 1 to 1. 


1 


Add. 


1 


Add. 




Deduct. 


if. of 


Deduct. 


n 




n 




1 




« 




1 


.6851 


31 


38.4629 


J0062 


31 


6.9321 


2 


1.4074 


32 


40.29^ 


2 


.0247 


32 


6.3210 


8 


2.1666 


88 


42J666 


8 


.0655 


33 


6.7222 


4 


2.9629 


84 


44.0740 


4 


.0988 


34 


7.1358 


6 


3.7962 


35 


46.0185 


5 


.1543 


35 


7.5617 


& 


4.6666 


36 


48.0000 


6 


.2222 


86 


8.0000 


7 


5.6740 


37 


50.0185 


7 


.8025 


37 


8.4506 


8 


6.6186 


38 


62.0740 


8 


.3961 


38 


a9136 


9 


7.6000 


89 


541666 


9 


.6000 


39 


9.3888 


10 


8.6185 


40 


56.2962 


10 


.6173 


40 


9.8765 


11 


9.5740 


41 


58.4629 


11 


.7469 


41 


10.3765 


12 


10.6666 


^ 


60.6666 


12 


.8889 


42 


10.8888 


18 


11.7962 


43 


62.9074 


13 


1.0432 


43 


11.4135 


14 


12.9629 


44 


65.1861 


14 


1.2099 


44 


11.9506 


15 


141666 


45 


67.5000 


15 


1.3889 


46 


12.5000 


16 


15.4074 


46 


69.8518 


16 


1.5802 


46 


13.0617 


17 


16.6851 


47 


72.2407 


17 


1.7839 


47 


13.6368 


18 


18.0000 


48 


74.6666 


18 


2.0000 


48 


14.2222 


19 


19.3518 


^ 


77.1296 


19 


2.2284 


4» 


148209 


20 


20.7407 


60 


79.6296 


20 


2.4691 


50 


16.4321 


21 


22.1666 


61 


82.1666 


21 


2.7222 


51 


16.0556 


22 


23.^296 


52 


84.7407 


22 


2.9876 


52 


16.6913 


23 


26.1296 


63 


87.3518 


23 


3.2664 


63 


17.3395 


24 


26.6666 


54 


9aoooo 


24 


3.5665 


54 


18.0000 


25 


28.2407 


56 


92.6861 


26 


3.8580 


56 


18.6728 


26 


29.8518 


66 


95.4074 


26 


41728 


66 


19.3680 


27 


31.5000 


67 


98.1666 


27 


4.6000 


67 


20.0665 


28 


33.1851 


68 


100.9629 


28 


48396 


58 


20.7654 


29 


34.9074 


69 


103.7962 


29 


6.1913 


69 


21.4876 


30 


36.6666 


60 


106.6666 


30 


5.6565 


60 


22.2222 



( CIXT. ) 

BASE 36— SLOPE 1^ to 1. 





Add. 




Add. 


Dif. of 


Deduct 


Dif. of 
Uti. 


Deduct. 


1 


.6898 


31 


^9120 


.0077 


81 


7.4161 


2 


1.4259 


32 


46.0370 


2 


.0309 


32 


7.9012 


3 


2.2063 


33 


47.2063 


3 


.0694 


33 


8.4027 


4 


3.0370 


34 


49.4259 


4 


.1234 


34 


8.9197 


5 


3.9120 


36 


61.6898 


5 


.1928 


36 


9.4621 


6 


4.8333 


36 


64.0000 


6 


.2778 


36 


10.0000 


7 


5.8009 


37 


56.3564 


7 


.3781 


37 


10.6632 


8 


6.8148 


38 


68.7592 


8 


.^38 


88 


11.1419 


9 


7.8750 


39 


61.2083 


9 


.6250 


39 


11.7861 


10 


8.9814 


40 


63.7037 


10 


.7716 


40 


12.3456 


11 


10.13^ 


41 


66.2463 


11 


.9336 


41 


12.9706 


la 


11.3333 


42 


68.8333 


12 


1.1111 


42 


13.6111 


13 


12.6787 


43 


71.4676 


18 


1.3040 


48 


14.2680 


14 


13.8703 


44 


74.1481 


14 


1.5123 


44 


14.9382 


15 


16.2063 


46 


76.8750 


15 


1.7361 


46 


15.6260 


16 


166925 


46 


79.6481 


16 


1.9763 


46 


16.3271 


17 


18.0231 


47 


82.4675 


17 


2.2299 


47 


17.0447 


18 


19.6000 


48 


86.3333 


18 


2.5000 


48 


17.7777 


19 


21.0231 


49 


88.2463 


19 


2.7865 


49 


18.5262 


20 


22.5925 


60 


91.2037 


20 


3.0864 


60 


19.2901 


21 


24.2083 


51 


942083 


21 


8.4028 


61 


20.0694 


22 


26.8703 


62 


97.26^4 


22 


8.7346 


52 


20.8641 


28 


27.5787 


63 


100.3564 


23 


4.0818 


68 


21.6743 


24 


29.3333 


64 


103.6000 


24 


4.4444 


64 


22.5000 


25 


31.1342 


65 


106.6898 


25 


4.8228 


65 


23.3410 


26 


32.9614 


56 


109.9269 


26 


6.2160 


66 


24.1975 


27 


34.8750 


67 


113.2083 


27 


5.6250 


57 


25.0694 


28 


36.8148 


58 


116.6370 


28 


6.0494 


68 


25.9567 


29 


38.8009 


59 


119.9120 


29 


6.4891 


69 


268696 


SO 


40.8333 


60 


123.3333 


30 


69444 


60 


27.7777 







( olxvi. ) 




BASE 36— SLOPE 1^ to 1. 


} 

m 
1 


Add. 


1 


Add. 


1 


Dednet. 


l« 


Deduct. 


.6844 


31 


47.3611 


.0092 


81 


8.8961 


2 


1.4444 


82 


^.7777 


2 


.0370 


32 


9.4815 


8 


2.2500 


83 


52.2600 


3 


.0833 


83 


10.0833 


4 


3.1111 


34 


54.7777 


4 


.1480 


84 


10.7037 


6 


4.0277 


85 


57.3611 


6 


.2313 


86 


11.3^25 


6 


5.0000 


86 


60.0000 


6 


.3333 


86 


12.0000 


7 


6.0277 


37 


62.6944 


7 


.4537 


87 


12.6759 


8 


7.1111 


38 


65.4444 


8 


.5926 


38 


13.3708 


9 


8.2500 


39 


68.2500 


9 


.7500 


89 


14.0833 


10 


9.4444 


40 


71.1111 


10 


.9269 


40 


14.8148 


11 


10.6944 


41 


740277 


11 


1.1208 


41 


15.5648 


12 


12.0000 


^ 


77.0000 


12 


1.3333 


^ 


16.3333 


13 


13.3611 


43 


80.0277 


13 


1.6648 


43 


17.1202 


14 


14.7777 


44 


83.1111 


14 


1.8148 


44 


17.9269 


15 


16.2500 


45 


86.2500 


15 


2.0633 


45 


18.7500 


16 


17.7777 


46 


89.4444 


16 


2.3704 


46 


19.5925 


17 


19.3611 


47 


92.6944 


17 


2.6769 


47 


20.4587 


18 


21.0000 


48 


96.0000 


18 


8.0000 


48 


21.3333 


19 


22.6944 


49 


99.3611 


19 


8.3426 


« 


22.2314 


20 


24.4444 


50 


102.7777 


20 


8.7037 


50 


28.1481 


21 


26.2500 


61 


106.2500 


21 


4.0883 


61 


240888 


22 


28.1111 


52 


109.7777 


22 


4.4815 


62 


25.0370 


23 


30.0277 


53 


113.3611 


23 


4.8981 


63 


26.0092 


24 


32.0000 


64 


117.0000 


24 


5.3333 


54 


27.0000 


26 


34.0277 


55 


120.6944 


26 


6.7869 


66 


28.0092 


26 


36.1111 


66 


124.4444 


26 


6.2592 


56 


29.0370 


27 


38.2500 


57 


128.2600 


27 


6.7500 


67 


80.0833 


28 


40.4444 


58 


132.1111 


28 


7.2692 


58 


31.1481 


29 


•^.6944 


59 


136.0277 


29 


7.7869 


59 


82.2314 


30 45.0000 


60 


140.0000 


30 


8.3833 


60 


3.33333 





( clxvii. ) 


BASE 36— SLOPE If to 1. 


t 

s 


Add. 


n 


Add. 


Dif. of 
Ht). 


Dednct. 


gas 


Dednct. 


1 


.6990 


31 


51.8101 


1 


.0108 


31 


10.3811 


2 


1.4629 


32 


54.5185 


2 


.0432 


32 


11.0617 


» 


2.2916 


33 


57.2916 


3 


.0972 


33 


11.7638 


4 


3.1852 


34 


60.1296 


4 


.1728 


34 


12.4876 


6 


4.1435 


35 


63.0325 


5 


.2700 


35 


13.2330 


6 


5.1666 


36 


66.0000 


6 


.3889 


36 


14.0000 


7 


&2456 


37 


69.0625 


7 


.^93 


37 


147885 


8 


7.4074 


38 


72.1296 


8 


.6913 


38 


15.5987 


9 


8.6250 


30 


75.2916 


9 


.8760 


39 


16.4305 


10 


9.9074 


40 


78.5185 


10 


1.0802 


40 


17.2839 


11 


11.2456 


41 


81.8101 


11 


1.3071 


41 


18.1689 


12 


12.6666 


42 


85.1666 


12 


1.6555 


42 


19.0666 


13 


141435 


43 


88.5879 


13 


1.8256 


43 


19.9747 


14 


15.6852 


44 


92.0740 


14 


2.1 173 


44 


20.9135 


15 


17.2916 


45 


95.6250 


15 


2.4305 


45 


21.8750 


16 


18.9629 


46 


99.2407 


16 


2.7664 


46 


22.8680 


17 


20.6990 


47 


102.9212 


17 


3.1219 


47 


23.8626 


18 


22.6000 


48 


106.6666 


18 


3.5000 


48 


24.8888 


19 


243657 


49 


110.^68 


19 


3.8997 


4ld 


25.9367 


2U 


2&29e2 


50 


1143518 


20 


43210 


60 


27.00,61 


21 


28.2916 


51 


118.2916 


21 


47639 


61 


28.0972 


22 


30.3518 


52 


122.2962 


22 


5.2284 


52 


29.2098 


£3 


32.4768 


53 


126.3657 


23 


5.7145 


63 


30.3441 


24 


34.6666 


54 


130.5000 


24 


&2222 


64 


31.6000 


25 


36.9212 


55 


1346990 


25 


6.7616 


55 


32.0774 


26 


39.2407 


56 


138.9(^ 


26 


7.3024 


66 


33.8765 


27 


41.6250 


67 


143.2916 


27 


7.8760 


57 


36.0972 


28 


440740 


58 


147.6852 


28 


8.4691 


56 


36.3394 


29 


46.5879 


59 


152.1435 


29 


9.08^ 


59 


37.6033 


ao 


49.16661 60 


156.6666 


30 


9.7222 


60 


38.8888 



( clxviii. ) 


BASE 86— SLOPE 2 to 1. 


eight 


Add. 


1 


Add. 


if. of 
ite. 


Deduct. 


if. of 


Deduct. 


n 
1 




s 

81 




1 




|Q 




.7037 


66.2592 


.0123 


31 


11.8642 


2 


1.4814 


82 


59.2692 


2 


.0494 


82 


12.6419 


3 


2.3333 


33 


62.33:38 


8 


.1111 


33 


13.4444 


4 


3.2592 


84 


66.4814 


4 


.1976 


34 


14.2716 


5 


4.2692 


35 


68.7037 


5 


.3086 


35 


16.1234 


6 


6.8333 


36 


72.0000 


6 


4444. 


36 


16.0000 


7 


6.4814 


37 


75.3703 


7 


.6049 


37 


16.9012 


8 


7.7037 


38 


78.8148 


8 


.7901 


88 


17.8271 


9 


9.0000 


39 


82.3333 


9 


1.0000 


39 


18.7777 


10 


10.3703 


40 


85.9269 


10 


1.2346 


40 


19.7630 


11 


11.8148 


41 


89.5926 


11 


1.4938 


41 


20.7680 


12 


13.3333 


42 


98.8333 


12 


1.7778 


42 


21.7777 


13 


14.9269 


43 


97.1481 


13 


2.0864 


43 


22.8271 


14 


16.5926 


44 


101.0370 


14 


2.4197 


44 


23.9012 


15 


18.3833 


45 


106.0000 


15 


2.7778 


45 


26.0000 


16 


20.1481 


46 


109.0370 


16 


3.1606 


46 


26.1284 


17 


22.0370 


47 


113.1481 


17 


3.6679 


47 


27.2716 


18 


24.0000 


48 


117.8333 


18 


4.0000 


48 


28.4444 


19 


26.0370 


49 


121.6925 


19 


4.4668 


49 


29.6420 


20 


28.1481 


60 


126.9269 


20 


4.9382 


50 


80.8642 


21 


30.3333 


51 


180.3333 


21 


5.4444 


61 


82.1111 


22 


32.5925 


62 


134.8148 


22 


6.9758 


62 


33.3827 


23 


34.9269 


63 


139.3703 


23 


6.5309 


63 


34.6790 


24 


37.3338 


54 


144.0000 


24 


7.1111 


64 


36.0000 


26 


39.8148 


55 


148.7087 


26 


7.7160 


65 . 


37.8456 


26 


42.3708 


66 


163.4814 


26 


8.3457 


66 : 


38.7160 


27 


45.0000 


57 


168.3383 


27 


9.0000 


67 ' 


40.1111 


28 


47.7037 


58 


163.2692 


28 


9.6790 


58 ' 


41.5808 


29 


50.4814 


59 


163.2592 


29 


10.3827 59 1- 


12.9758 


80 


53.3333 


60 173.8833 | 


80 11.1111 ' 60 [ 


144444 



(clxi..) 


BASE 36— SLOPE ^ to I. 


1 


Add. 


1 


Add. 


I 


Deduct. 




Deduct. 


1 


.7129 


31 


66.1674 


1 


.0154 


31 


148302 


2 


1.6185 


32 


68.7407 


2 


.0617 


32 


15.8016 


3 


2.4166 


33 


72.4166 


3 


.1389 


33 


168055 


4 


3.4OT4 


34 


76.1851 


4 


.2468 


34 


17.8396 


S 


44907 


36 


80.0462 


6 


.3867 


35 


18.9043 


6 


5.6666 


36 


84.0000 


6 


.5565 


36 


20.0000 


7 


6.9361 


37 


88.0462 


7 


.7561 


37 


21.1265 


8 


8.2962 


38 


92.1851 


8 


.9676 


38 


22.2839 


9 


9.7500 


39 


96.4166 


9 


1.2500 


39 


23.4722 


10 


11.2962 


40 


100.7407 


10 


1.5432 


40 


24.6913 


11 


12.9351 


41 


105.1674 


11 


1.8673 


41 


25.9413 


12 


14.6666 


42 


109.6666 


12 


2.2222 


42 


27.2222 


13 


16.4907 


43 


114.2686 


13 


2.6080 


43 


28.6360 


14 


18.4074 


44 


118.9629 


14 


3.0247 


44 


29.8765 


15 


20.4166 


45 


123.7600 


15 


3.4722 


43 


31.2500 


16 


22.6186 


46 


128.6296 


16 


3.9506 


46 


326543 


17 


24.7129 


47 


133.6018 


17 


4.4599 


47 


34.0895 


18 


27.0000 


48 


138.6666 


18 


5.0000 


48 


36.6556 


19 


29.3796 


49 


143.8240 


19 


6.5710 


49 


37.0324 


20 


31.8518 


60 


149.0740 


20 


6.1728 


60 


38.6802 


21 


34.4166 


51 


164.4166 


21 


68055 


31 


40.1388 


22 


J7.0740 


52 


159.8518 


22 


7.4691 


52 


41.7283 


23 


39.8240 


63 


163.3796 


23 


8.1636 


53 


43.3487 


24 


12.6666 


54 


171.0000 


24 


8.8889 


54 


45.0000 


23 


15.6018 


55 


176.7129 


26 


9.6466 


55 46.6820 || 


<M 


OCtCMUt 


w 


lOORIOj; 


Oft 


flAWl 


Kfl 


taxinn 11 





( clxx. ) 




BASE 36— SLOPE 3 to 1. 


1 


Add. 


•a 

•s 


Add. 


if. of 

Ht8. 


Deduct. 


if. of 
Elts. 


Deduct. 


SB 
1 




93 


* 


1 




Q 




.7222 


31 


74.0655 


.0185 


31 


17.7963 


2 


1.5656 


32 


78.2222 


2 


.0740 


32 


18.9630 


3 


2.6000 


33 


82.6000 


3 


.1667 


33 


20.1666 


4 


3.5666 


34 


86.8888 


4 


.2963 


34 


21.4074 


5 


47222 


35 


91.3888 


6 


.4628 


35 


22.6851 


6 


&0000 


36 


96.0000 


6 


.6667 


36 


24.0000 


7 


7.3888 


37 


100.7222 


7 


.9074 


37 


25.3518 


8 


8.8888 


38 


105.5655 


8 


1.1852 


38 


26.7407 


9 


10.6000 


39 


110.6000 


9 


1.5000 


39 


28.1666 


10 


12.2222 


40 


115.6565 


10 


1.8518 


40 


29.6296 


11 


14.0565 


41 


120.7222 


11 


2.2407 


41 


31.1296 


12 


16.0000 


42 


126.0000 


12 


2.6667 


42 


32.6666 


13 


18.0556 


43 


131.3888 


13 


3.1296 


43 


34.2407 


14 


20.2222 


44 


136.8888 


14 


3.6296 


44 


35.8518 


16 


22.5000 


45 


142.5000 


15 


4.1667 


45 


37.6000 


16 


24.8888 


46 


148.2222 


18 


4.7407 


46 


39.1851 


17 


27.3888 


47 


154.0555 


17 


5.3618 


47 


40.9078 


18 


30.0000 


48 


160.0000 


18 


6.0000 


48 


^.6666 


19 


32.7222 


49 


166.0555 


19 


6.6852 


49 


44.4629 


20 


35.5555 


50 


172.2222 


20 


7.4074 


50 


46.2963 


21 


38.6000 


61 


178.5000 


21 


8.1667 


61 


49.1666 


22 


41.6655 


62 


184.8888 


22 


8.9629 


52 


50.0740 


23 


44.7222 


63 


191.3888 


23 


9.7962 


63 


62.0185 


24 


48.0000 


64 


198.0000 


24 


10.6667 


54 


54.0000 


25 


51.3888 


55 


204.7222 


25 


11.5741 


56 


56.0184 


26 


54.8888 


66 


211.5555 


26 


12.5184 


56 


68.0740 


27 


58.5000 


57 


218.5000 


27 


13.5000 


57 


60.1666 


28 


62.2222 


58 


226.5555 


28 


14.5186 


58 


62.2962 


29 


66.0555 


59 


232.7222 


29 


15.5741 


59 


64.4629 


30 


70.0000 


60 


240.0000 


30 16.6667 1 60 


66.6666 







( 


ci jxi. ) 










TABLE OF BASES. 








I 


Z 


3 


4 


S 




1 


.0185 


0370 


.0556 


.0741 


.0920 




2 


.0370 


0741 


.1111 


.1481 


.1852 




3 


.0556 


nil 


.1667 


.2222 


.2778 




4 


.0741 


1481 


.2222 


.2963 


.3704 




5 


.0926 


1852 


.2778 


.3704 


.4630 




6 


.1111 


2222 


.3333 


.4444 


.6566 




7 


.1296 


2593 


.3889 


.5186 


.6481 




8 


.1481 


2963 


.4444 


.5926 


.7407 




9 


.1667 


3333 


.5000 


.6687 


.8333 




10 


.1852 


3704 


.5556 


.7407 


.9269 




11 


.2047 


4074 


.6111 


.8148 


1.0186 




12 


.2222 


4444 


.6667 




1.1111 




13 


.2407 


4816 


.7222 


!9630 


1.2037 




14 


.2583 


5185 


.7778 


1.0370 


1.2963 




15 


.2778 


3556 


.8333 


1.1111 


1.3889 




16 


.2963 


5926 


.8889 


1.1852 


1.4816 




17 


.3148 


6296 


.9444 


1.2592 


1.5741 




18 


.3333 


6667 


1.0000 


1.3334 


1.6667 




19 


.3519 


7037 


1.05.56 


1.4074 


1.7692 




20 


.3704 


7407 


1.1111 


1.4814 


1.8518 




21 


.3889 


7777 


1.1687 


1.5556 


1.9444 




22 


.4074 


8148 


1.2222 


1.6206 


2.0370 




23 


.4259 


8519 


1.2778 


1.7037 


2.1297 




24 


.4444 


8889 


1.3333 


1.7778 


9.9-m. 




25 


.4630 


9259 


1.3889 


1.8519 


2.3148 





( clxxii. ) 
TABLE OF BASES. 





1 


2 


3 


4 . 


S 


31 


.5741 


1.1481 


1.7222 


2.2963 


2.8704 


32 


.5926 


1.1852 


1.7778 


2.3704 


2.9630 


S3 


.6111 


1.2222 


1.8333 


2.4444 


3.0556 


34 


.6296 


1.2593 


1.8889 


2.5185 


3.1481 


35 


.6481 


1.2963 


1.9444 


2.5926 


3.2407 


36 


.6667 


1.3333 


2.0000 


2.6667 


3.3333 


37 


.6852 


1.3704 


2.0556 


2.7407 


3.4259 


38 


.7037 


1.4074 


2.1111 


2.8148 


3.5185 


39 


.7222 


1.4444 


2.1667 


2.8889 


3.6111 


40 


.7407 


1.4815 


2.2222 


2.9630 


3.7037 


41 


.7593 


1.5185 


2.2778 


3.0870 


3.7963 


42 


.7778, 


1.5556 


2.3333 


3.1111 


3.8889 


43 


.7963 


1.5926 


2.3889 


3.1852 


3.9815 


44 


.8148 


1.6296 


2.4444 


3.2592 


4.0741 


45 


.8333 


1.6667 


2.5000 


3.3334 


4.1667 


46 


.8519 


1.7037 


2.5556 


3.4074 


42592 


47 


.8704 


1.7407 


-2.6111 


3.4814 


43518 


48 


.8889 


1.7777 


2.0667 


3.5555 


44444 


49 


.9074 


1.8148 


2.7222 


3.6296 


4.5370 


50 


.9259 


1.8519 


2.7778 


3.7037 


46297 


51 


.9444 


1.8889 


2.8333 


3.7778 


4.7222 


52 


.9630 


1.9269 


2.8889 


3.8519 


48148 


53 


.9815 


1.9630 


2.9444 


3.9260 


4.9074 


54 


1.0000 


2.0000 


3.0000 


4.0000 


5.0000 


55 


1.0185 


2.0370 


3.0556 


4.0741 


5.0926 


56 


1.0370 


2.0741 


3.1111 


4.1481 


5.18^ 


57 


1.0556 


2.1111 


3.1067 


4.2222 


5.2777 


58 


1.0741 


2.1481 


3.2222 


42963 


5.3704 


59 


1.0926 


2.1852 


3.2778 


4.3704 


5.4630 


60 


1.1111 


2.2222 


3.3333 


4.4444 


5.5555 



A, 



" cd = he^ht at the other. 

" bd = length of prismoid. 

" abed ~ the vertical section through the 
the direction of its length. 
This figure is that view commonly represente 
aad by reference to which all the calculatia 



174 

A section is a plot of a succession of levels taken along a 

given line of country (see Fig. 2), shewing the comparative 

levels of any two I points^ and also the distance between 

them. 

Fig. 2. 




The line C D represents the datum line on a horizontal 
plane^ assumed according to circumstances^ for the purpose 
of shewing the comparative levels of any given places; 
the line AB is the gradient determined to be worked on 
by the engineer, which likewise has its comparative levels 
in reference to the datum line C D ; the Ime E P is the 
surface of the coimtry, consisting of hills and valleys on 
the site of the line of the proposed work. According as the 
gradient AB is below or above the surface of the country 
EF, as shewn on the Fig. 2, the whole quantity of earthwork 
is required to be taken from the space E ABF, if the former 
case ; or it is necessary to fill up with earth, imposed on 
the surface E F, a quantity sufficient to form the embank- 
ment indicated on the section by the space AEBF. To 
render the computation as simple as possible, and to reduce 
it within the known rules of geometry, at every variation 
of level there are perpendiculars let fall (or raised, as the 
case may be) to the gradient AB, thereby making the 
section a series of simple figures, and similar to that shewn 
as Fig. 1, which is the prismoid. The ordinary way of 
dividing the section is by squaring up lines with the datum 
CD ; but if the gradient is not a level, but takes a direction 
AB^, then the figure is no longer a rectangular prismoid 
ABFE, (see Fig. 3) ; but a prismoid AB' FE where neither 
the angles EAB* or AB^F axe right angles. 



^e prismoid due to the sor&ce EF is greater in that 
case than in the rectangnhtr caae : I will shew the difference 
Airther on, and merely remark here, that in practice, 
supposing snch difference existed in the prismoids, FG 
represents the height due to a rectangnlar prismoid, while 
PB' is the actual height measnred j the error is due to the 
difference of the heights FB' and FG. Yet the steepest 
gradient rarely exceeds 1 in 100; bo that when the per- 
pendicular height FGwith the gradient is 1, the distance 
GB' = -L, then the height PB' due to the perpendicular 
to the base line is ^i + JL,; the error is so trifling that it 
need never he regarded in calculations of this kind for 
practical purposes. 

When the sections are prepared, the slope of the aides 
is determined upon according to the local and contingent 
circumstances, and the base of the work is also given. 
These are the only other quantities involved which are not 
represented on the section; for on inspection of Pig, 1, it 
is seen that the area ABCD of the transverse section 
consists of a parallelogram of the base B Cj and height B I, 
and two triangles of the height BI, and abase AI due to 
the slcpe. Therefore, by knowing the heights on the section, 
and the lengths, the cubic contents can be determined, 

The rule given in books on geometry and o^ 
works is as follows : — 

To the area of the two ends add four timt 
section, and divide the sum by one-sixth of I 
prismoid ; or (algebraically expressed) 



( 



176 

Let H = height of one end. 
" A = height of the other end. 
" L = length. 
" B = given base. 
" S = given slope. 

Then (HB + ffS + AB+ A^S + 4[(?J*) B + (iLlir)s]x| 

is the cubic content of the prismoid. 
When developed^ this equation becomes 

[HB + ff S + AB+ A'S+ 2BH + 2BA + ff S^ 2HA+ A- S] + ^ ; 

and when reduced^ it becomes 

r(H+A)B (H« + H*+^5Sl T i.- ^ i. 

I' — 2*-^ + ^ 3 — '""^ J L = cubic contents. 

Now, this rule being given, in order to establish its 
accuracy I will prove it by the method of fluxions, and 
take it m the most simple form, when H = o, or the surface 
of the ground at one end intersects the base (see Fig. 4). 

^'^- ^' Let AB be the gra- 

dient,* and AC the 
surface, and the Ex- 
pressions in algebra 
^ bethesameasbefore.f 

Let any variable area at DE be equal to y, and AD, 
a variable length, be equal to x. Now ya? = fluxion of cube. 




* In all cases where the gradient is mentioned, the formation level 
is intended, which is generally 2 feet 6 inches lower than the level of 
the rails. The word '' gradient" signifies a certain rise above, or foil 
below, the horizontal line in some given length. 

t These expressions will always be used throughout the whole 
essay ; and, occcasionally, b, h, s, I are used, instead of B, H, S, L. 



177 
and as AB : AD : : BC : DE, or DE = ^: and the area 

then 18 ( -J- + -p~) = y ; therefore —^ — + — p- = 
fluxion of cube : the fluent of this is -j-j- -^ -j-p- = cube ; 
and when x = 1, cube = "a" * ~ ~ ABC. Now, by 
deducting the priam ADE from ABC, we can find 
the priamoid EDBC. Introducing the other height 
H, EDBC = i (»H , £|!) _ (»_» _ I») ,. BatDB 
being giTen=L, x=l — Lj so, asa? : A :: L : (H — A) j tlierefore 
, - At. = i — L, and ! - L ( Ji-J. Substitute 
this value of L and 'a? in tte former equation, 

1-7* 3 ' H— * ^3 3 ' H^ 

Bednce ttas, L (-^ ♦ -5- - -y - -j) - g^ -d 
tterefore L (»- '!^) * -J '^') = ». But 
, H"— y-lH + AjxCH— 4)ilience^^ = H + S. And 
H<_tf=(H'*HJt»-) (H— 4)1 therefore ^■-H'tH»+A>. 
Consequently ['ili«» * (iSjJUia 
which is the same as the ordinary rule 

For the benelil of those who do not 
fluiiooal process, I subjoin a geometric n 



178 

Let fig. 5 be a prismoid in geometric perspective^ and the 
slopes of the sides be assumed of equal ratio. 

Fi«. 6. 




I 



The soKd figure is ABCDEFGH. The base of the work 
being BC, draw perpendiculars to BC through B and C, 
and complete the solid IBCKLMGFby the intersecting 
vertical planes on B G and C F. The prismoid is then 
divided into three figures^ namely^ the two equal soUds 
AIBGMH, and DKCFLE, which are frustra of pyra- 
nuds ; and the middle solid IBCKLMGF^ which is a 
frustrum of a rectangular wedge. If the analysis is carried 
on in the same manner as the former^ by assuming the 
surface and the base to meet^ as in fig. 4, the results will 
be obtained in a more simple form. 

Using the same notation as before, we have — 

h = height. 
I = length. 
s = ratio of slope, 
b = base. 



i 



179 



%.6- 




I^ore 6 represents 
the isoiQj^iic p»^6(s 
tiTe of sQidi soEdy whoe 
aQ the pboies meet at 
FE. 

First^ the wedge 

HBCGEFis= ^' 
Seo(nidfy,t]iepjnanid8AHBFaadDGCEue= ^ 

Therrfove the som is ^^ + ^-^ = cabe. This equation 

bemg the same as prodnoed by the fluent in the fluxional 
method, of conrae^ mnst bnng oat the same resnlts, 
which, when detennined for a prismoidj becomes 

j-6(H+j^ ^ |.(H*+H* + *«)]'*«• Mr, MacncB, in his 

tables, has also proved the role in a different manner to 
that followed hoe, though not in so concise a form of 
investigation, and produces at once the role ordinarily given. 
f^ Having now shewn the principle of the piismoidal 
formula, I shall proceed to compare it with the ordinary 
method of measuring work by contractors. There are two^ 
ways in use. One is to take the mean height of a section 
(supposed to be subdivided), and, finding the area, then 
to multiply by the length for the cube, or algebraically, 

5±* =meaii height; [?i^+ (H'.aHA^y^^j^ j^^^ 

Compare this with the prismoid, and we find that the 
quantities that differ are those due to the sides, therefore 

4 — 3 

3A2 + 12j? = 4ff + 4 HA + 4 As consequently 
± 12 ^ = ff — 2HA + tf == (H — A)*. But as 
(H — A)^ is always a positive quantity, the sign of the 
other term of the equation lolLJMidHIBlis^i so that 




180 

X = j2 — "= *^® quantity required to be added to the 

area of the mean height^ shewing that that process gives 
the quantity less thw the true one. The second method 
used by contractors is to find the areas of the ends of a 
section^ and take the mean area^ and multiply by the length 
as before for the cube^ or algebraically expressed^ 

|- BH.H»S.B*.yS j , i = eube = [B(Ha) . I (ff .^]/. 

Comparing this with the prismoid^ it is the side quantities 
that differ again; hence 5!l^ ± a? = 5Lt-5^ 

8H' + 3A* ± 6^ = 2H« + 2HA + 2A»; consequently 
ff— 2HA + A' =+ 6ar=(H — A)»; and (H — A)« being 
always positive^ x is always negative by transposition; 

so that (lLzi*)Lf = quantity to be deducted from the 

mean area founds proving that this method is always 
in excess^ and^ comparing the two methods together, 
the latter is twice as much in excess as the former is 
in deficiency. As a numerical illustration may shew the 
comparative results more clearly^ I will suppose two heights^ 
50 feet and 80 feet^ and a length 500 feet^ on a base 35 
feet^ with a slope of 2 to 1. 

By the prismoidal formula^ 
80 X 50 =- 1500 = HA 



30 X 30 
60 X 50 


E> 


900 
2500 




a)80= H + A 
40 


3)4900 

1633i 
2 


_ H« + HS + A« 


35 -B 


3 

= 8 


1400 = b("'" 

2 


32661 





so 

30 





181 




3266| 




600 = L 




3333000 
333 


.,{ 


3)2333333 


9) 777777.7 


86419.7 cubic yardj. 
By mean heights — 



2)80 

lo . 85 - 1400 = fflifiS 

40 X 40 = 1600 « 2 = 3200 = EtiSf 



4600- 



(H + A) B (H 4- fl)" « 
2 "^ 4 

500 
3)2300000 



27 ri: 

t 9) 766666| »^^ 

85185.1 c^'^J 

Mean Heights = 85185.^; 
Prismoid = 86419.^ 



Deficiency 







182 

By mean areas — 
50 X 35 = 1750 1750 = BH 

50 X 50 = 2500 X 2 = 5000 = H»« 
50 X 35 = 1050 1050 = BA 

30 X 30 = 900 X 2=1800 = /*'» 



2)9600 



4800 = 5i^ + 0^1^* 
500 



3)2400000 
27 " 

9) 800000 



{ 



88888.8 cubic yards. 



Mean areas = 88888.8 cubic yards. 
Prismoid = 86419.7 " 



Excess = 2469.1 cubic yards. 

This single calculation shews that the error is very great 
for such an ordinary length as 500 feet. A contractor 
who uses these methods may, therefore, tender for a greater 
quantity than is true by the method of mean areas ; and 
by the method of mean heights, he may tender for a less 
quantity than true : so that he incurs the danger of two 
extremes — in the first case being liable to be refused, 
because his figures exceed the estimate of the engineer, 
based on the prismoidal formula ; and of being disadvan- 
tageously met by other contractors, who may have availed 
themselves of the true method ; or otherwise, if he should 
tender according to the results derived in the second case, 
and he should be accepted, he would find that, on his 
work being measured, he would have to perform more 
than he contracted for^ at his own cost; besides the general 



183 



effect the error in either case would have upon the schedule 
of prices. On the other hand^ it is necessary that the 
resident engineer should be careful not to allow the use 
of the method of mean areas^ as the contractor would 
thereby receive more money than his work performed would 
warranty the contractor's interest evidently preventing him 
firom using the other way. The consideration of the above 
I therefore earnestly recommend to both parties^ that they 
may do justice to each other as well as to themselves. 

I have hitherto assumed that all the prismoidal sections 
are rectangular^ although in page 175^ I alluded to cases 
where they are otherwise^ owing to the inclination of the 
gradient. Though at first sight the two cases may appear 
to be alike^ yet the foUowing demonstration will explain 
the difference. For simplicity^ I will assume the section 
as in Fig. 7^ which represents an ordinarv small cuttings 
commencing at the surface of the ground at A, the summit 
being at B^ and the termination at C. Here A C is the gra- 

Rg. 7. dient; and ABC is the 

surface of the ground; 
BD is a height which is 
common to both pris- 
moids ABD and DBC. 
When AC is horizontal^ 
then BD is perpendicular 
to it; but if the gradient takes another direction Kcy 
being the same lengthy and the surface ABC being equal 
to KbCy then 6rf, the perpendicular to Ac, is equal to 
B D ; and i e, a vertical line through the point A, meeting 
Ac in e, is greater than bd; for the angle bde being a 
right angle, A e is the hypothenuse of the triangle bde^ 
and 6 e is also common to the prismoids A & e and ebc ; 
but the lengths A c and A C are equal, and B D and b e 
being given, the mean areas can be found, which multiplied 
by the lengthy will give the cubic contents. Therefore, 
because 6 e is greater than B D^ the solid A & c is greater 




184 

than the solid ABC. This result shews that the method 
of squaring the divisions from the base line is not perfectly 
accurate, ^ough, for the reasons previously given, suffi- 
ciently so for practice. 

It not unfrequently hi^peus that, in an irregular 
country, the ground not only varies considerably in the 
direction of the line of works, but also transversdy, so as 
to affect the quantity to be measured in a very sensible 
manner. In cases of this description, it is necessary to 
take cross sections of the country at every sub-division of 
the ordinary section, and sometimes oftener; on which 
occasion the section must be further sub-divided to inter- 
sect the cross sections. 
In the subjoined Fig. 8, the line AB is the gradient 
Fig. 8. 



depressed below the sui&ce of the country, and the section 
is sub-divided at various points a, a, a, into simple figures ; 
and the lines ba, ba, ba, represent the cross sections at the 
points a, a, a, having reference to perpendiculars at those 
places, or toa hwizoatal datum line like an ordinarysection. 



185 




The most simple case of the prismoid with a cross sec- 
tion is^ when two cross levels^ some measured distance 
from each other^ produce the same inclination^ for then it 
is sufficient to find the areas in the same manner as for 
the common prismoid; but when two levels vary, then it 
becomes necessary to determine the law of the variation, 
and find the sohd by the fiuxional process, which is the 
most ready method. 

In Kg. 9, let the area ABCD represent a transverse 
^^S< 9« section, where AD is 

the natural slope of the 
ground. Draw the hori- 
zontal line LH, passing 
through the centre (E) 
of the section, EP is 
the height of the section 
at the centre of the base. 
Let r be the ratio of the natural slope, when the vertical 
height is = 1. The area ABCD consists of the triangle 
ABN, the trapezium NBCO, and the triangle OCD. 
Because EP is the mean between NB and OC, EP x BC 

=area of NB CO. The triangle OCD = (EH— PC.) x ^ ; 

BN 
and the triangle ABN = (EK— BP) x — . 

The slopes of the railway being assumed equal on 
both sides, therefore EL = EG. 
As EG: GI::r: 1 ::EH : HD. 
AsGH = EH— EG : HD :: * : 1. 

Hence EH = HD x r; therefore HD = M . 

T 

AlsoEH— EG = HD x s; therefore, HD = EH -EG 

AndEH5 = EHr— EGr 

EH (r— «)=EGr; therefore, EH = EG -I-, being the 

width on one side of the centre. 

A A 



186 

Afl EL : LM :: r : 1 :: EK : KA. 
As KL =EL— EK : KA :: 8 : 1. 

Hence EK == KA x r; therefore KA = — 

Alao EL— EK = KA x «; therefore, KA = 51^=5? 

And EK* = EL x r— EK x r 

EK (r + «) = ELr; therefore, EK = EL-I-, or 

EK = EG -^, being the width on the other side of the 

centre. 
Now since EG = |- + **> 

WehaTeEH= *1^ {^^) 

And EK= *^ (-^) 

Also O C = EP * LC = A . I. = 2-^ 

And BN = EP-«/=A- i = ^^*. 

From the above equations we derive the whole area ABCD 
as follows 



TheteiangleOCD= (^-±|Ai) (?^^) ^^_| (?4±i). 

The triangle ABN = (AlfA.^) (^^^) l-^-| (^-A^^ 

Trapezium NBCO = A A, 

Consequently, the whole area ABCD is — 

When the slopes of the sides and the natural slope of 
the ground are given, the area may be found speedily, if 
they can be referred to the accompanying table. The 
formula being simplified first into — 

b± ^ 2j^b ^^^IizPyr ^^\) ; which, again reduced, 



187 



is T -^ ^^ [It^*] ' ^*' ''y expounding this 
form, we have** + 4,^,AU2 (r'^^) 6>^,y ^^ g^^ 

and making J^. =X; i + /j-^^,, = Y; —J—^^=Z; 



ff X + BHY + B'Z = area. 



Slope 1 to 1. 



Slope litoL. 



Slope 2 to 1. 



R 


X 


5 


1.0416 


10 


1.0101 


15 


1.0044 


20 


1.0025 


25 


1.0016 


30 


1.0011 


35 


1.0008 


40 


1.0006 


45 


1.0005 


50 


1.0004 



1.0416 
1.0101 
1.0044 
1.0025 
1.0016 
1.001 1 
1.0008 
1.0006 
1.0005 
1.0004 



R X 



.01083 

.00252 

.00111 

.00062 

.00040 

.00028 

.00020 35 

.OOOI5U0 

.OOOI2U5 

.oooiolso 



5 
10 
15 
20 
25 
30 



1.6666 
1.5345 
1.5151 
1.5085 
1.5054 
1.5037 
1.5027 
1.5021 

1.5017 
1.5013 



1.1111 
1.0230 
1.0101 
1.0056 
1.0036 
1.0025 
1.0018 
1.0013 
1.0011 
1.0008 



R X 



.01673 

.00384 
.00168 
.00094 
.00060 



5 
10 
15 
20 
25 



00042 30 



.00030 



35 



00023 40 



.00018 
.00015 



45 
50 



2.3810 
2.0833 
2.0363 
2.0202 
2.0129 
2.0090 
2.0065 
2.0050 
2.0040 
2.0032 



I.O9O6 
r04l6 
1.0181 
1.0101 
1.0064 
1.0045 
1.0032 
1.0025 
1.0020 
1.0016 



.02381 
.00521 
.00226 
.00126 
.00080 
.00056 
.00041 
.00031 
.00024 
.00020 



To render this table more comprehensible^ let us assume 
a numerical example- 
Let the height be 80 feet") And by referring to the table. 



« 



34 feetf slope 2 to 1, and opposite the 
2 to 1 Rvalue of R, we find the respec- 

35 to 1 3 tive coefficients X, Y, Z. 
X X H2 = 900 X 2.0202 = 1818.18 

Y X BH=1020 X 1.0101 = 1030.30 
Z X B' = 1156 X .00126= 1.45 



base 
slope 
Natural slope 



9)2849.93=area in feet. 

316.66=square yards area. 

If the area had been computed, supposing the ground 
to be level, then — 

aa2 



188 

BH = 30 X 34 = 1020 
H»x *= 30 X 30x2= 1800 



9)2820 



313.33 square yards. 

Therefore^ this example shews^ that if the natural slope of 
the ground had not been considered^ 3.33 square yards 
would have been lost in the area. As the solid figure is 
not altered in regard to its principles, by the consideration 
of the natural ground slope, the cubic contents can be 
found by modifying the formula for the simple prismoid. 
For, using the same notation as before, and introducing the 
second height, the solid is 

= L X [B^Z + 2 + 3 J' 

and if the mean areas of the ends be found, assuming them 
to be calculated by the express formula for sloping ground, 

the correct nile is [ A±J! _ (H-j^)^X j ^ j, = the solid, 



when A and a are the areas of the ends. 




Let this be illustrated 


numerically : 




H = 30 feet 








A = 20 feet 




In the table 


we find — 


B = 34 feet 




X- 


2.0129 


S = 2 to 1 




Y = 


1.0064 


R = 25 to 1 




z = 


.0008 


L -400 feet 








ff X = 900 


X 


2.0129 == 


1811.61 


A^X = 400 


X 


2.0129 - 


805.16 


B(H+A)Y= 50 X 


34 


X 1.0064 - 


1710.88 


2 B' Z - 1156 "^ 


2 


X .0008 = 


1.85 




2)4329.50 


Mean area 


— 


2164.75 



(H— A)'X X(30-20) ' 



2164.75 = mean area of the two ends. 

33.54 ^ correction. 

2131.21 =^ true mean area of prismoid. 

400 = length. 

f 9)852484.00 = contents in cubic feet. 

i 3) 94720.44 



31573.48 = cubic yards in solid. 
Comparing this with the simple prismoid — 
S H= = 900 X 2 - 1800 
a A= = 400 X 2 - 800 
B(H+A}= 50 ^ 34- 1700 

2)4300 



(H; 


-hf. 


= correction = 
2160 

33.33 - 


2150 -«^ean area of the two ends. 
'^' = 33.33 

mean area. 

correction. 




2116.66 - 
400 = 


true mean area of the prismoid. 
length. 




9)846664.00 - 

3) Mors.rr 


contents in cubic feet. 



31357.92 = cubic yards in solid. 
By prismoid with sloping ground = 31573.'; 
By " level do. = 31357.i 

Therefore, by this example, it appears that 21f 
yards would be lost by neglecting to estimate for t 
slope of 1^ gronnd. 



190 




Wlien the natural slope of the ground continually varies^ 
in order to determine the cube accurately, it is necessary 
to ascertain the law of the variation of the slope ; which 
process, if it were possible in practice, would involve so 
many calculations, as to render it tedious and inapplicable 
for practical men ; but it is a very near approximation to 
the real value, if the natural slope of the ground is sup- 
posed to increase directly in proportion to the distances. 

Let Fig. 10 represent a transverse section of an uniform 
Fig. 10. rising section. J£ we 

suppose the base EC 
to intersect a horizontal 
ground surface trans- 
versely, but as the sec- 
tion advances, the cross 
sections assume the 
figures EBCF, GBCH, 
ABCD, whose natural 
ground slopes vary in proportion to the distance from the 
intersecting point, through which BC passes; that is, if the 
section EBCF is at half the distance at which ABCD is 
situated, the ratio of 01 to IF is half the ratio of ML to 
LD; and also if the ratio of NK to KH of the section 
GBCH (whose centre point N is midway between M and 
O) is half the sum, or the mean between the ratio ML to 
LD, and O I to IF. Then if these ratios are given, it is 
easy to determine the cube very nearly. For this solid 
may be considered as only another modification of the 
prismoid ; and whereas the increments of the heights and ^ 
slopes are uniformly proportional, we shall find that the 
same relation exists between the prismoid and the rules 
for mean heights or areas, as has been shown in former 
examples. Therefore, making use of the former notation 
and example, and adding a new ratio of slope r = 15 to 1 
due to one of the ends, we have, by using the tables, the 
following method to find the mean area : — 



191 



L = 400 feet. 



H = 80 R = 25 B = 34 
h=20 r = 15 S=2 

H + h = 30+20 = 25 = A = mean height. 

2 2 

R+_r ^ 25_ii5 _ 20 ^ ^ ^ ^ej„ slope. 

2 2 

In the table we find the following coefficients : — 



Due to R = 25. 
X = 2.0129 

Y = 1.0064 

Z = .0008 z = .00226 

H2 X = 900 X 2.0129 = 1811.61 
'BHY = 34 X 30 X 1.0064 = 1026.53 



Due to r = 15. 
X = 2.0362 

y = 1.0181 

z = 



y 

z 



Due to r = 20. 
2.0202 
1.0101 
.00126 



B^Z =1156x 



.0008 = 



.92 



Px = 400 
Bhy = 34 X 20 
B z = 1156 



(A) 

2.0362 = 814.48 

1.0181 = 692.31 

.00226= 2.61 



Area of 

2889.06 5 greater 
c end. 



(B) 



Area of 
1509.40 { ^^^ 



A»a? = 625 X 2.0202 = 1262.62 
BAy = 34x25 X 1.0101 = 858.58 
B2 r = 1156 X .00126 = 1.45 



Area of 



(C) 2122.65x4=8490.60 J ™f- 

c sec. 



Calculation carried forward 
2139.84 
400 



6)12839.06 



Mean area 2139.84 



c 8)855986.00 
C 9)285312.00 



31701.33 cube yards — true contents of solid; 



192 

Compare this with the mean height method^ we have 

2122.65 Area (C) 

400 cube yds. 

By prismoid 81701.83 

3)849060.00 By mean heights 81446.66 

27' 

9)283020.00 Deficiency 254.66 



I 



81446.66 cube yards. 
And comparing it with the method of mean areas^ we 
have— 

2839.06 = greater end (A) 
1509.40 = lesser end (B) 

2)4348.46 

2174.23 = mean area of the two ends. 
400 



cube yds. 
3)869692.00 By prismoid 31701.33 

27^ By mean areas 32210.81 

9)289897.33 

— . Excess 509.48 

32210.81 cube yards. 

I>i£Perence between mean heights and mean areas 764.15 
cubic yards. 

These results shew that, though not exactly correct, the 
method by the prismoidal formiila is nearer the truth than 
either of the other two, and the method is nearer and 
nearer the usual result, as the given variable ratios of the 
slopes approach to equality. 

In case the formation of the surface assumes a curve, the 
nearest method that can be used in practice is by laying 
down such a straight line as will give the same area, and 
determine the mean slope of the particular cross section : 
this being repeated as often as may be necessary, the 
remaining process is the same as before to find the cube. 



193 




If inagiven cross section there may be found two slopes 
(rednced from a cmre or otherwise) whose intersection can 
be conveniently determined, within two vertical lines fall- 
ing on the extremities of the base; then the base being 
subdivided by a line perpendicular to it, from the intersec- 
tion of the slopes, two separate calculiUions may be made 
of the respective areas, whose sum is equal to the whole 
transverse section. Thus, in fig. 11, ABCD is a cross 

^. !!• section, whose sur&ce 

AED is composed of 
two slopes AE, ED 
meeting at E. The 
base BC is dividedinto 
twolengUis, BH, HC; 
and the area ABCD 
can be computed by 
finding the sum of 

ABHEandEHCD. 

Now, as the mean height EH, at the intersection of the 
slopes = H ; and BH, the part of the base cut oflF, can be 
measured = b ; then substituting this height for the usual 

height, and b for ?, the general expression for each figure 

becomes (t±ML- *) x (*^^) * (?Iff^) *. 
This equation reduced to its simplest terms is 
(ftA+Z^ -+ — ). The sign plus being used in the 

r . . ft' 

coefficient 77+7, and the sign minus in the term + ^ 

when the ground/a/& towards the centre, and contrariwise 
when the ground rises. 

* 

When the nature of the ground is such that it becomes 
necessary to use two different slopes on opposite sides of 



r + « 



194 



an embankment or cuttings it is sometimes convenieat to 
compute the contents by taking half the snm of the cubes 
produced^ supposing the section to be calculated with each 
slope separately. But^ if the tables used admit di it, the 
simplest way is to take the mean between the two given 

slopes, and work out the cubes as usual. For as ^— 
is the area of the one side. H^ (5jL£2 is the area of the 

2 

two sides. But in the case of inclining ground^ the slopes 
must be separated^ as the equation of each side has a co- 
efficient depending on the particular slope; therefore, the 

rule given for the last case applies here, making A = ? 

or half the base, if the inclination of the groimd is uniform. 
It sometimes happens, after the estimates are completed, 
that, owing to the insufficiency or improper disposal of 
materials, according to the existing gradients and other 
causes, the engineer alters the gradient, and thereby ren- 
ders it necessary to undertake another series of calculations. 
The easiest way is to make use of the prismoidal tables 
with the amended heights ; but as it may be sometimes 
useful to know the area due to the diiference of the heights^ 
I will subjoin a solution of this problem. 

Let Fig. 12 represent a transverse section. ABCD is 
l^iR- 12. the original area; and 

BF is an increase in 
depth, the base and 
slopes remaining the 
same. It is obvious 
that the additional area 
is the figure BFEC, 
and twice CEGD. If 
the parallelogram CEGH be completed, and HK drawn 
perpendicular, and BC FE be produced to meet HK, 
then the rectangular parallelogram CEKI is equal to 




195 

CEGH ; CEGD 18 equal to CEGH — the right-angled 
triangle GHD. But DGis to GH as the ratio of the 
slope; and CI is equal to the snm of the side base due 
to the height + DG, or CI is the base dne to the increased 
height. Therefore, in notation, making d = increase of 
depth, we haveBE=A(*and'xCEK;i =2rf (A + rf) s— (fa; 
therefore the increase of area i& d [b -^ {2h + cQ s] ; 
or, if d represented the reduction of depth, the area to be 
deducted is d [4+ (2A— d) s]. 

The next case which comes under our notice is, when 
in any given section the dopes do not continue uniformly 
{torn the gradient to the sur&ce of the ground, but are 
interrupted by a bench being introdueed at a given point 
of the height. Let fig. 13 represent such a cutting. AB 
Fig. 13. isthebasejHGthesur- 

. face; FC DE benches 
introduced. Here, by 
producing the slope 
BD to meet the sur- 
face in I, the psrallelo- 
gram IGED is the ad- 
ditional area to each side of the ori^al section. Therefore, 
if the height <rfDE firom ABbeglTen=A', and the breadth 
DF=ft,then2 (H — A') b = area to be added to the ormnal 
section. K these benches extend in parallel planes wim the 
gradient, then the additional soUd is ( H^*-3ft')6L 

The last case but one I shall observe is, when the widths 
in a plan are given, to find the cubic contents. 





196 

In the diagram Pig. U, let A BCD represent a plan of a 

^*^' ^^' prismoid; and let the dotted 

TTj lines represent the respec- 
tive sections at the ends. 
That is, let ABEP be the 
section at AB ; andCDHG 
be the section at CD. Pro- 
duce FE HG both ways, 
and let fall on them the 
perpendiculars, completing 
the rectangles A I K B, 
C MLD. Insert W for A B, 
and w for CD, and use in 
other respects the previous 
notation. 

It appears by the construction that 

WH— H2S = areaof ABEForAIKB— 2 AIP; 

wh~hU = area of CDHG or CMLD— 2 CMG. 

But W = 2 H« + B, therefore H = 5L=M. 
^ffll j^ W^— WB H3 S = '^''— 2 WB H- B^ 

Therefore, the area AEBF = W^z:5!, and substituting 

half the width and half the base for their respective wholes, 

we have ^^^=2? = area ABEP. 
s 

Treating the other area in a similar manner, we find 
CDGH = ^^-^—^i therefore, the solid contents of 
A BP.PaTTT^n -c r^g^!=g . ^^^' . W^2\fw^vP^ w -x L. 

\. s s s J^e 

This equation reduced becomes [(WliZi^+u^ — ?' "I x L 

It is to be observed that this equation is simpler than that 
given in the terms of heights ; for as « and B are known, 



197 

th^e i« only one unknown terra, from which a constant 
quantity is deducted, to find the mean area. 

It only remains now to find the cubic contents of a 
prismoid with sloping ground, when the extreme widths 
are given. Let us assume a case where the slope is uniform 
throughout. 

Fig. 15, 




Let Fig. 15 represent the proposed prismoid whose cube 
is required. Let NK be a line parallel to the base EF, 
intersecting the surface AB in I, the vertical centre of 
the base, and cutting the slopes in M and N ; and let fall 
the perpendiculars AE, B£, through A and B on the line 
NK. By a former caae we know thatIL= IN (— ^1, and 

IK=IN (-^^); therefore LK=IN(^). AlsoAB^Wis 
the hypotbenuse of a triangle whose base and perpend icula r 
are in the ratio of r to 1: theiefore AB ^ W = LK i"^^.) 

IJet cLK ='W, when c is the coefficient ■ t - 
,j,_BiiHS therefore W - tilH-S, J^ 
merefore H - '^fc^ 



198 



By the fonner case the area vas found to foe 

If half the base and half the width be given^ instead of 
the whole base and widths and the value of H be substi- 
tuted, we have area A = 

and inserting the value of c in the equation — 

Now as H = W (;7v^=^) — 7 $ a^d it may be 

sometimes desirable to know the height in the middle, the 
subjoined table is constructed in order to facilitate the 
calculations. 



The column X is computed by the formula ( 



r^—s^ 



s (r+H) 



); 



and the column C by the formula = ( . ^ ) 





Slope 1 to 1. 




Slope Ij 


to 1. 


b: 


X 


C 


R 


X 


C 


6 


.92308 


.94137 


5 


.58334 


.59490 


10 


.98022 


.98487 


10 


.64523 


.63368 


15 


.99115 


.99337 


15 


.65708 


.65855 


20 


.99502 


.99625 


20 


.66195 


.66278 


25 


.99680 


.99760 


25 


.66379 


.66423 


30 


.99780 


.99835 


30 


.66461 


.66498 


35 


.99838 


.99877 


35 


.66490 


.66518 


40 


.99875 


.99907 


40 


,66531 


.66553 


45 


.99900 


.99926 


45 


.66560 


M576 


50 


.99922 


.99942 


50 


.66581 


.66595 



Slope 2 to 1. 



R 

5 

10 
15 
20 
25 
30 
35 
40 
45 
50 I 



.40385 
.47524 
.48893 

.49377 
.49601 
.49723 
.49796 
.49843 
.49876 
.49900 



C 



.41185 
.47762 
.49002 

.49439 
.49641 
.49751 

.49817 
.49860 
.49890 
.49910 



I will illustrate this table by a numerical example : 
Given, The natural slope R = 15 to 1. 
Half the base = 17 feet. 
Half the width = 108 feet. 
The slope being = 2 to 1. 



Here X = 0.M89S; tbnefiHc. 
+ 108 X lOS X 0.48893 = 5702.88 

— 17 K 17 « J = 144.50 

Feet 5558.38 meu uea. 
Nov C = 0.49002} therefore, 

+ 108 - 0.49002 =- 52^ 

— 17-1 = 8.50 

Feet 44.42 heiglit in the middle. 
Having ibond the height, let us prove the preceding 
calculation of the area hy the table given in page 187, in 
the terms of the heights and base. 

Here X = 3.0362 j Y = 1.0181; Z = .00226; therefore, 
44.42 X 44.42 x 2.0362 = 4017.70 
44.42 X 34 X 1.0181 -= 1537.62 
34 X 34 X 0.00226 = 2.61 

Feet 5557.93. mean area; 
the difieience being only such as may be due to a 
trifling error in the last place of decimals in the table. 
The only difference in fiuding the solid in this case and 
the last will be by affixing to the term in which W and 
V? are found, the coefficients X due to the sloping ground, 
instead of the coefficient ~ > the sohd then becomes 



[.(^^±W^)_^"]. 



RULES RELATING TO THE PRISMOIDAL 

FORMULA. 
To avoid the necessity of referring to the essay "■ — '- 
late cubic contents, I subjoin the following — 

BTTLEB FOB PRACTICAL CASES. 

1. — To find the contents of a primurid ( 
cross section is horizontal}, 
B H + H' s From the mean aiea of the two en 



200 

BA + A? S one-Bixth of the product of the slope^ multi- 
2 plied by the square of the diflference of the 
(H— .*)» S heights. This difiference multiplied by the 
6 length will give the contents. 
2. — To find the area of a transverse section of a 
railway [when the cross section is horizontal). 
B H + H' S Multiply the base by the height, and add 
to the product the product of the slope and 
the square of the height. 
3. — To find the area of a transverse section {when 
the cross section is sloping). 

Let 1 to r be the ratio of the sloping grouud, 
r being the horizontal length ; and let 1 to # 
be the ratio of the slope of the cutting; then 

making X = -3^^,, Y = 4 + -4^x 

Z = 4 (y^ —s^Y "^^^ ^^^ "^^ ^® found by 

H'X + HBY adding together the product of the square of 
+ B'Z the height by X, the product of the base by 
the height by Y, and the product of the 
square of the base by L. 

4. — To find the cubic contents of a prismoid (the 

cross section being sloping). 

A + g Prom the mean area of the two ends deduct 

2 one-sixth of the product of the coefficient x 

{U — h)^x (found by rule 3), and the square of the diflFer- 

6 ence of the heights. The difference multiplied 

^ L by the len^h gives the contents. 

5. — To find the area due to an alteration in the level 
of abase or formation line. 

first Case. — If the depth is increased: 

Multiply the sum of twice the original height, 

rf[B+(2H+rf)«]and the increase of depth, by the slope; to 

this add the base, and multiply the sum by the 

increase of depth for the increase of area. 



201 

Second Case. — If the depth is diminished : 
Multiply the difference of twice the original 
height, and the diminution of depth, by the 
d[B+(2ft-d)5]slope; to this add the base, and multiply the 
sum by the diminution of depth; the result 
gives the area to be deducted. 
6. — To find the additional soUd by having benches 
on each side (parallel with the base line), 
(H+A — 2A*)6L From the sum of the heights deduct twice 
2 the height of the bench above the base ; mul- 
tiply the difference by half the width of the 
bench, and by the length ; the product is the 
contents. 
7 — When the half-undths of the cuttings or embank- 
ments at the edge of the slopes against the surface 
are given, to find the area. 
W^—B^ Divide the difference of the squares of half 

s the width and half the base by the slope ; 

the result is the area of the section. 
8. — When the tvidths are given (as above), to find 
the cubic contents. 
W+Ww-^u^ Add together the square of half the greater 
3 « width, the square of half the lesser width, and 
the product of both half widths; divide this 
__ B^ sum by 3, and from the quotient deduct the 

s square of half the base. This difference divi- 

ded by the slope, and multiplied by the length, 
will give the cubic contents. 
9. — When the tvidtlis are given, to find the area 
{w?ien the cross section is sloping). 

Let 1 to r be the ratio of the sloping ground; 
and 1 to * be the ratio of the slopes of the 

cutting ; make x = ^~" . Then the pro- 
duct of the square of half the width, by the 
B2 coefficient x, less by the quotient of the square 
^ Y of half the base divided by the slope, is the area, 

B B 



202 

10. — To find the height, given the width {when the 
cross section between is sloping). 

Wc — - Make C = ^5=^=, > ^^®^ *^® ground 

is level, the coefficient C becomes = 1 

then, by dividing half the base by the slope, 
and deducting the quotient from the product 
of half the width, and the coefficient C, the 
result is the height in the middle. 

11, — To find the width when the height is given {the 

cross section of the ground being sloping) . 

To the height add the quotient of half the 

B hnsG divided by the slope, and divide the sum 

7 by the coefficient C as before found; the 

"C result is the width. 

12. — When the uoidths are given, to find the cubic 

contents {the cross section being sloping ground). 

x{W^+Ww+w^ ) Proceed the same as in rule 8, substitut- 

* ing for coefficient j the coefficient -; when 

_ B- .r is found by the formula in rule 9. The result 

7 is the cubic contents. 





EXPLANATION 



OF THE USB OP 



TABLES OF EARTHWORK. 



In preparing these tables^ the object I had in view was 
to do away with the necessity of turning over many pages 
and of referring to the intersection of two columns of given 
heights to find the tabular number^ and also to serve other 
useful purposes ; and I thought that if I could reduce the 
table required for any given slope and base to one page^ I 
should materially simplify the operations^ though appa- 
rently at the expense of a few extiui figures. The tables of 
Mr. Mac Neill are constructed partly for given bases and 
slopes, and partly for abase 1, with a series of slopes. The 
former tables, by reference to the intersection of the given 
heights, at once produce the mpan area of the prismoid ; 
but these tables are very limited in extent of base and 
slope. The latter tables, by a similar reference, give a 
number in one column to be multiplied by the given base, 
and this is added to a number found in the same line of 
intersection in a second column. The sum is the mean 
area required. I here subjoin an example ; the two heights 

B b2 



J 



204 

being 18 and 38, and the slope being 1 to 1, &nd the basi 

32 feet :— 

By Second Serws of Tables. 

In base 1, slope 1 to 1, opposite IS and under 38 ve find- 
Tabular niunber . 1.037 
Multiply by base . . 32 



Add tabular number . . 30.272 

Mean area . . 63.456 
In using either series of tables it is necessary to turn 
over a few pages eonatantly, which renders the application 
tedious, though very far preferable to calculating by the 
formula. These tables, moreover, do not admit of any 
appbcation of them except for the purpose of cubing the 
pnamoid. 

Other tables have been published for computing the 
quantities for any given base, but they are not more ad- 
vantageous than Mac Neill's, but are more economical in 
cost, and have been serviceable, owing to the present 
scarcity of Mac Neill's work. 

Mac Neill's tables are computed by the formula 
/H + S\ - /H"+ HA + ft'\ , . , .„ . 

^_ — 2 — J B + ^^^ — - — 3 ) a, which will give the 

mean area in feet, and divided by 27 gives bis tabular 

number in his first series of tables at each intersection 

:n heights. In his second series, he has divided 

ular numbers into two columns at each mtersec- 

' the heights, the first bemg computed by the 

5^ and the second by the term ( ^i ^ «• 

dimensions are measured in feet. 



205 

As I wish to save as much as possible the intersecting 
method, which, in a work of the extent I have brought 
out, would have been abnost impracticable on account of 
the labour and expense attending it, and being desirous of 
applying the same tables to other purposes, I sought and 

found the formula — ^ + 2 — 6 

well adapted for the purpose. This is obviously the mean 
area of the two ends, with the correction applied to render 
the prismoid true, the particular investigation of which will 
be found at page 180. By the use of this formula I keep 
the heights independent of each other, except in the third 
term, when their difference is required. Therefore in 
determining the values of the terms for any series of bases, 
slopes, and heights, the two first terms are exactly similar, 
and can be classed in one table ; and the third term, de- 
pending upon the difference of heights, can be classed in 
a second table ; and this term is also common in all the 
given bases for the same slope. But in the tables I have 
reprinted it in every base, so as to save the trouble of 
turning to another page. Therefore the tables are com- 

BH + H«S 
puted for the first two columns by the term — gj , 

and the table of differences by the term y^ In order 

to use the tables the following rules are given : — 

To Calculate the Oube. 
Look in the table of the given base and slope opposite 
the two given heights, for the numbers under the columns 
headed '^ add,'^ and add these together : this corresponds 
with the mean area of the two ends. Then look in the 
columns headed '' deduct,^' for the number opposite the 
difference of the heights : by deducting this from the sum 
previously obtained, the mean area of the cube will be 
found, which, multiplied by the length, wiU give the cubic 
contents in yards. All the dimensions must be given in 



206 

feet. Take^ for an example^ the dimensions before given 
in illustration of Mac Neill's tables — 

Base 32, Slope 1 to 1. 

Col. Add. 

Opposite 18 we find . . . . 16.6666 

Ditto 38 ditto . • . . 49.2592 



65.9258 
Col. Deduct. 
Difierence of heights being 20, oppo- 
site 20 we find .... 2.4691 



Mean area . . . 63.4567 
It is obvious, in comparing this method with Mr. Mac 
Neill's, that the operation of adding and svhtracting, re- 
quired by the former method, is much less liable to error 
than the operation of multiplying and adding required by 
the latter. Besides, the two added numbers at once give 
an approximate result, which may frequently su£5[ce for a 
rough estimate, without subtracting at all. 

I now give a form which will illustrate the calculation 
of a cutting or embankment complete. I used these 
tables for some years on the Eastern Counties Railway ; 
and for the convenience of preserving the calculations, I 
had a number of books ruled in this form, in the manner 
of a levelling book. .And the whole process admits of being 
checked in a similar manner, except the multiplication : 
for the sum of the columns of addition in columns 1 and 2 
should equal the sum of the various sums column 3 ; and 
the sum of the deductions, column 4, deducted from the 
sum in column 3, should correspond with the sum of the 
differences in column 5. The lengths are written in 
column 6; and the product of columns 5 and 6 are 
written in column 7 : the sum of these products will give 
the quantity in the cutting or embankment. In columns 
8 and 9 are written the respective heights at either end ; 
and column 10 gives the difference of the heights. 









Base 


32, Slope 1 to 1. 












1 2 


3 




5 


6 


7 


8 9 


10 




Add (ogelhcr. 


Bum. 






Unglh. 


Frodd. 


S'H 


DH. 01 












BfBrt. 


lu.rd.. 


hBlihu. 




0,0000 2.6666 


2.6666 


.0968 


2.5678 


120 


308 


01 4 


4 




2.6666 


5.9259 


8.5925 


.0988 


8.4937 


260 


2208 


4 ! 8 






5.9529 


14.2222 


20.1481 


.3951 


19.7530 


340 


6716 


8 


16 


8 




14.2222 


24.8888 


S9.1110 


.3951 


38.7159 


280 


10840 


16 


24 


8 




24.88BB 


37.9259 


52,8147 


.3951 


61.4196 


670 


41151 


24 


32 






14.2222 


11.5259 


25.7481 


2.0000 


23.7481 


8?0 


20661 


16 


14 


18 




11.5259 


6.0555 


16.5814 


.5000 


16.0814 


460 


7397 


14 


7 


9 




5.055S 


1.9444 


6.9999 


.0988 


6.9011 


330 


3277 


7 


3 






1.94441 O.O0O0 


1.9444 


.0555 


1.8889 


130 


246 


31 


3 








Total 


ubic yards— 




91804 









In the above form there IS no necessity to repeat bo many 
of the figures, for it is obvious that the same height is 
common to both adjoining prismoids, and the same tabular 
nmnber will apply, except in such cases where a jump or 
sudden break in the ground occurs at the same place as 
illustrated in line six of the cdculation ; except in such a 
case, it would be sufficient merely to write the numbers in 
columns 2 and 9^ omitting those in columns 1 and 8. There- 
fore for each priamoid two tabular numbers are required 
to be abstracted, both found in tlie same page ; the one for 
the last height of the prismoid, and the other for the 
dijference of the he^hts. The number due to the first 
ka.ght of the prismoid is the same as that due to the last 
height of the prismoid immediately preceding, so that it 
will have been already entered in the form. The quickest 
way, having first transferred all the measurements to the 
form, is to extract tlie tabular numbers at once s< 
and then proceed to the addition, subtraction, and 
plication of the whole. After a little practice th( 
will soon become habituated to the tables. In j 
three places of decimals are amply accurate. Th< 
the tables, and of the above form, will be foiu 



208 

simple, if, instead of the lengths of each prismoid being 
unequal, the given cutting or embankment was divided 
into any number of equal parts; let the heights be 
measured as usual, and the tabular numbers entered in 
the form ; for then, in order to find the cube, sum up 
column 8, from which deduct the sum of column 4, and 
multiply by the whole length of the cutting or embank- 
ment, and divide the product by the number of prismoids 
of equal length : the quotient is the required cube. This 
last method wiU probably reduce the calculation of a long 
cutting or embankment to one operation of multiplication 
{* ^!S^^^?i^^^ ^^ several, thereby considerably reducing the 
J5Mffl9rt)|^error, and enabling the whole of the work to be 
easily ch)^cked. 

To Calculate the Area of the Section. 

Multiply the tabular number in the columns headed 
" add,^' opposite any given height in feet, by 6 j and the 
product is the area in square yards. Hence in the 
preceding form any of the numbers in columns 1 and 2 
may be taken to find the area of the section ; for instance, 
24.8888 X 6 = 149.3328 = area in yards, corresponding 
with 24 feet height in column 8. The numbers in columns 
1 and 2 themselves are half the cubic contents, in yards, 
due to the section of a prism 1 foot long. 

To Calculate the Area of the Sides. 

Multiply the tabular number in the columns headed 
^^ deduct,'* opposite any given height in feet, by 6; and 
the product is the cubic contents, in yards, due to 1 foot 
of length: this again multiplied by 3 will give the area of 
the sides in yards. 

The preceding form, therefore, enables us to take a 
memorandum of the section, the base and slope and situa- 
tion on the section being indicated at the head of the 
form, the lengths and heights being given in columns 6, 
8, 9 of the form ; also, to produce the sectional area at any 
given place, or the area of the sides, or the cube. 



209 

As it may be required to know the area of any section^ 
supposing that the base is made accidentally narrower 
or broader^ or to compute the cubic contents consequent 
upon such alteration^ I subjoin a table of bases^ 1, 2, 
Sy 4iy 5. By the use of this table^ suppose that it were 
required to compute the cubic contents of a cutting with 
base 18^ which is not in the table ; then compute as if 
for base 20^ and deduct firom each calculation of the 
mean area the sum of the two numbers due to the differ- 
ence of the bases (20 — 18 = 2) opposite the given heights 
in the tables of bases: the difference is the true mean area. 
To find the area of any given section for an altered base^ 
deduct or add the number opposite the given height in the 
column of bases which represents the decrease or increase^ 
from the tabular number of the given height in the 
original base ; six times the result is the true area. To 
find the area of the middle piece, multiply the number 
opposite the given height in base 1 by the given base ; 
six times the result is the area. And to find the cubic 
contents of the middle piece add together the corresponding 
numbers in base 1 due to the given heights, and multiply 
by the length, and by the given base : the result is the 
cube. 



210 

Having thus explained the use of the tables, 1 now 
proceed to give an explanation of a scale for measuring 
earthwork, which I have successfully used on 
the Eastern Counties Bailway. The vertical 
and horizontal scales, and also the hase and 
slope, must be previously determined for each 
scale, and then a set can be formed embnunng 
the requiredslopesoftherailway. My scales 
were made four in number, for base 84-; 
slopes,], Ij, Ij, and 2 tol; and on the back 
of each was properly described the slope, 
base, and vertical and horizontal scales to 
whitJi they were applicable. All the dimen- 
sions were taken in yards lineal, superficial, 
or cubical, as required ; the application being 
precisely the same as the use of the tables, 
and the rules being the same to iind the area, 
using 2 as a coefficient instead of 6. The 
annexed diagram shews one of the scales for 
slope 1 to 1, reduced to one-half of its real 
size, being calculated for 20 feet to I inch 
vertical scale, and 2i chains to one inch 
horizontal scale. This diagram ia, therefore, 
applicable to 40 feet to 1 inch vertical scale, 
and 5 chains to 1 inch horizontal scale. But, 
owing to the fineness of the graduations, I 
would not advise, to insure accuracy, that 
the vertical scale be made less than 40 feet 
to 1 inch, and, generally, the lai^er the 
better. 20 feet to 1 inch vertical is a good 
working scale. The lengtlis can be made 
to suit c 



There is no necessity to measure the heights or lengths 
previously, and the same form must be used as before 
described. 



211 

To Measure a Cutting by the Scale. 

Apply the zero of the scale of aectional areas vertically to 
the gradient or fonnation lincj and read off where the sunace 
line intersects ; put this in column I or 3, as the case 
requires : then, at the smaller end of the priamoid, upon 
the scale of vCTtical yards, with lero on the surface hue, 
observe where the gradient intersects, then place the same 
point of intersection on the gradient at the other end, and 
read off above the zero on the scale of differential areas, 
where the surface line interaects ; put this in column 4, and 
then, having measured all the lengths by the scale of 
horizontal yards, proceed as in the use of the tables. To 
save time, it is desirable to take a pair of dividers, and 
mark off at each diviaiou the difference of the heights in 
succession, and then the differential scale above zero need 
only be applied. 

This method of computing cubic contents of cuttings 
and embankments is very expeditious vrith a little practice, 
and is quite as accurate, and generally more so, than the 
calculating by feet and the tables j because, in a working 
section, the points of intersection of the scale and surface 
line can be estimated readily by the eye ; but, in using an 
ordinary scale of equal parts, we are compelled to neglect 
the fractional parts of a foot. Should any of my refers 
wish to obtain these scales, they can easily be made, by 
giving a reasonable notice to Mr. Elliott, High Holbom. 
the rule for forming them is as follows : — 

To Calculate the Sectional Areas. 

The formula is M'ea = g '^ ^i therefore, wlien 

the area is graduated 10, 20, 30, 40, &e., in even 
aion, we have 11=+ ^ H = 2 A j and the corre 



hei^tisH=^2A + ^— fs;^^«^^ = 34- 



212 



'^^^ A ^ 34^ 



in yards, and « =3 to 1, the rule becomes H = 2A+--i^^ — 

J 3 f 3*X4>«4 

— 3t|t^ =^^"' (?)'—¥• To find the limit of the 

graduation^ (a) the greatest height the scale may be 
wanted for must be determined. The scale of differen- 

tial areas is computed from the formula D = -g— ; therefore 

the differences being graduated 1^ 2, B, 4^ &c.; 10^ 20^ 30^ 

&c. seriatim^ we have the corresponding height, H = ^^, 

If required for slope 2 to 1, the rule becomes 1.728 VS. 

From the preceding explanation it is evident that the 
particular vertical scale being determined, a set of scales 
for any number of bases can be easily made, as the only 
variable portion is that of the sectional areas. These scales 
are very convenient and portable, and can be frequently 
employed in the absence of tables, by being carried in the 
pocket, enclosed in a case. 





(214) 






AREAS OF LAND 


FOR CUrriNGS AND EMBANK- 




MENTS. 






SLOPE f to 1. 


SLOPE 1 to 1. 


Ht. 


Sides. 


Ht. 


Sides. 


Ht. 


Sides. 


Ut. 


Sides. 


1 


0000172 


31 


0005337 


1 


0000229 


31 


0007116 


2 10000345 


32 


0005510 


2 


0000459 


32 


0007346 


3 


0000518 


33 


0005682 


3 


0000689 


33 


0007576 


4 


0000689 


34 


0005854 


4 


0000918 


34 


0007805 


5 


bo00861 


35 


0006026 


5 


0001148 


36 


0008036 


6 


b001033 


36 


0006198 


6 


0001377 


36 


0008264 


7 


0001205 


37 


0006371 


7 


0001607 


37 


0008494 


8 


0001377 


38 


0006542 


8 


0001836 


38 


0008723 


9 


0001550 


39 


0006714 


9 


0002066 


39 


0008953 


10 


0001722 


40 


0006887 


10 


0002296 


40 


0009182 


11 


0001893 


41 


0007059 


11 


0002624 


41 


0009411 


12 


0002066 


ASt 


0007231 


12 


0002754 


42 


0009641 


13 


0002238 


43 


0007403 


13 


0002984 


43 


0009871 


14 


0002410 


44 


0007575 


14 


0003213 


44 


0010100 


15 


0002586 


45 


0007748 


16 


0003443 


46 


0010330 


16 


0002755 


46 


0007920 


16 


0003673 


46 


0010569 


17 


0002927 


47 


0008093 


17 


0003902 


47 


0010789 


18 


0003099 


48 


0008265 


18 


0004132 


48 


0011018 


19 


0003272 


49 


0008436 


19 


0004361 


49 


0011248 


20 


0003444 


50 


0008608 


20 


0004591 


50 


0011478 


21 


0003615 


51 


0008781 


21 


0004820 


51 


0011707 


22 


0003788 


52 


0008953 


22 


0005060 


52 


0011937 


23 


0008960, 


53 


0009125 


23 


0005280 


53 


0012167 


24 


0004132 


54 


0009297 


24 


0005509 


54 


0012396 


25 


0004304 


55 


0009469 


25 


0005739 


66 


0012626 


26 


0004476 


56 


0009642 


26 


0006968 


56 


0012866 


27 


0004649 


57 


0009813 


27 


0006198 


57 


0013085 


28 


0004821 


58 


0009986 


28 


0006427 


58 


0013314 


29 


00G4992 


59 


0010158 


29 


0006667 


59 


0013644 


30 


0005165 


60 


0010331 


30 


0006887 


60 


0018774 



(215) 

AREAS OF LAND FOR CUTTINGS AND EMBANK- 
MENTS. 



SLOPE 1^ to 


1. 


SLOPE H to 1. 


Ht. 


Sides. 


Ht. 


Sides. 


Ht. 


Sides. 


Ht. 


Sides. 


1 


0000287 


31 


0008895 


1 


0000345 


31 


0010674 


2 


0000574 


32 


0009182 


2 


0000689 


32 


0011019 


3 


0000861 


33 


0009469 


3 


0001033 


33 


0011363 


4 


0001148 


34 


0009756 


4 


0001377 


34 


0011707 


5 


0001435 


35 


0010043 


5 


0001722 


35 


0012052 


6 


0001722 


36 


0010330 


6 


0002066 


36 


0012396 


7 


0002009 


37 


0010617 


7 


0002410 


37 


0012740 


8 


0002296 


38 


0010904 


8 


0002776 


38 


0013085 


9 


0002583 


39 


0011191 


9 


0003099 


39 


0013429 


10 


0002869 


40 


0011478 


10 


0003445 


40 


0013774 


11 


0003156 


41 


0011765 


11 


0003788 


41 


0014118 


12 


0003443 


42 


0012052 


12 


0004132 


42 


0014463 


13 


0003733 


43 


0012339 


13 


0004476 


43 


0014807 


14 


0004017 


44 


0012626 


14 


0004821 


44 


0015161 


15 


0004304 


45 


0012913 


15 


0005165 


45 


0015496 


16 


0004591 


46 


0013200 


16 


0005510 


46 


0015840 


17 


0004878 


47 


0013487 


17 


0005864 


47 


0016164 


18 


0005165 


48 


0013774 


18 


0006198 


48 


0016529 


19 


0005452 


49 


0014061 


19 


0006542 


49 


0016873 


20 


0005739 


50 


0014348 


20 


0006887 


60 


0017217 


21 


0006026 


51 


0014635 


21 


0007231 


51 


0017561 


22 


0006313 


52 


0014922 


22 


0007676 


52 


0017906 


23 


0006600 


53 


0015209 


23 


0007920 


53 


0016250 


24 


0006887 


54 


0015496 


24 


0008265 


54 


0018694 


25 


0007174 


55 


0015783 


25 


0006608 


55 


0018939 


26 


0007461 


56 


0016070 


26 


0008963 


56 


0019283 


27 


0007748 


57 


0016367 


27 


0009297 


67 


0019627 


28 


0008035 


58 


0016644 


28 


0009642 


58 


0019972 


29 


0008322 


59 


0016931 


29 


0009985 


69 
60 


0020316 


30 


0008606 


60 


0017217 


30 


0000330 


0020661 



(216) 

ACRES OF LAND FOR CUTTINGS AND EMBANK. 

MENTS. 



SLOPE If to 1. 


SLOPE 2 to 1. 


Ht. 


Sides. 


Ht. 


SidM. 


Ht. 


Sides. 


Ht. 


Sides. 


1 


0000401 


31 


0012463 


1 


0000469 


31 


001^233 


2 


0000804 


32 


0012866 


2 


0000918 


32 


0014692 


8 


0001207 


33 


0013268 


3 


0001377 


33 


0016161 


4 


0001607 


34 


0013669 


4 


0001836 


34 


0016610 


5 


0002009 


36 


0014061 


6 


0002295 


36 


0016070 


6 


0002410 


36 


0014462 


6 


0002766 


36 


0016629 


7 


0002812 


37 


0014866 


7 


0003214 


37 


0016988 


8 


0003213 


38 


0016266 


8 


0003673 


38 


0017447 


9 


0003616 


39 


0016667 


9 


0004132 


39 


0017906 


10 


0004017 


40 


0016069 


10 


0004691 


40 


0018366 


11 


0004417 


41 


0016470 


11 


0006060 


41 


0018824 


12 


0004820 


42 


0016872 


12 


0005609 


42 


0019283 


13 


0006222 


43 


0017274 


13 


0006968 


43 


00197^ 


14 


0005623 


44 


0017676 


14 


0006^27 


44 


0020201 


16 


0006026 


46 


0018078 


16 


0006887 


46 


0020661 


16 


0006428 


46 


0018479 


16 


0007346 


46 


0021120 


17 


0006829 


47 


0018882 


17 


0007805 


47 


0021679 


18 


0007231 


48 


0019283 


18 


0008264 


48 


0022038 


19 


0007633 


49 


0019684 


19 


0008723 


49 


0022497 


20 


0008035 


60 


0020086 


20 


0009183 


60 


0022966 


21 


0008435 


61 


0020488 


21 


0009641 


61 


0023416 


22 


0008838 


62 


00-20890 


22 


0010100 


62 


0023874 


23 


0009240 


63 


0021292 


23 


0010559 


63 


0024333 


24 


0009641 


64 


0021693 


24 


0011018, 


54 


0024792 


26 


0010043 


66 


0022095 


26 


0011478 


66 


0026262 


26 


0010444 


66 


0022497 


26 


0011937 


66 


0026711 


27 


0010847 


67 


002289ft 


27 


0012396 


67 


0026170 


28 


0011248 


68 


0023299 


28 


0012865 


68 


0026329 


29 


0011649 


69 


0023702 


29 


0013314 


69 
60 


0027088 


30 


0012052 


60 


0024105 


30 


0013774 


0027648 



(217) 




AREAS OF LAND FOR CU'lTINGS AND EMBANK- 


MENTS. 


• 


SLOPE 2i to 1. 


SLOPE 3 to 1. 


Ht. 


Sides. 


Ht 


Sides. 


Ht. 




Sides. 


Ht. 


Sides. 


1 


0000574 


31 


0017790 


1 


0000690 


31 


0021348 


2 


0001148 


32 


0018364 


2 


0001378 


32 


0022038 


3 


0001722 


33 


0018938 


3 


0002066 


33 


0022726 


4 


0002296 


34 


0019512 


4 


0002764 


34 


0023414 


5 


0002870 


35 


0020086 


5 


0003444 


35 


0024104 


6 


0003444 


36 


0020660 


6 


0004132 


36 


0024792 


7 


0004018 


37 


0021234 


7 


0004820 


37 


0026480 


8 


0004592 


38 


0021808 


8 


0005511 


38 


0026170 


9 


0005166 


39 


0022382 


9 


0006200 


39 


0026858 


10 


0005738 


40 


0022956 


10 


0006890 


40 


0027548 


11 


0006312 


41 


0023530 


11 


0007576 


41 


0028236 


12 


0006886 


42 


0024104 


12 


0008264 


42 


0028926 


13 


0007460 


43 


0024678 


13 


0008952 


43 


0029614 


14 


0008034 


44 


0025252 


14 


00096^ 


44 


0030302 


15 


0008608 


45 


0025826 


15 


0010330 


46 


0030992 


16 


0009182 


46 


0026400 


16 


0011020 


46 


0031680 


17 


0009756 


47 


0026974 


17 


0011708 


47 


0032368 


18 


0010330 


48 


0027548 


18 


0012396 


48 


0033058 


19 


0010904 


49 


0028122 


19 


0013084 


49 


0033746 


20 


0011478 


50 


0028696 


20 


0013774 


50 


0034434 


21 


0012052 


51 


0029270 


21 


0014462 


51 


0035122 


22 


0012626 


52 


0029844 


22 


0015150 


52 


0036812 


23 


0013200 


53 


0030418 


23 


0016840 


53 


0036500 


24 


0013774 


54 


0030992 


24 


0016530 


64 


0037188 


25 


0014348 


55 


0031666 


25 


0017216 


55 


0037878 


26 


0014922 


56 


0032140 


26 


0017906 


56 


0038666 


27 


0016496 


57 


0032714 


27 


0018594 


67 


0039254 


28 


0016070 


58 


0033288 


28 


0019284 


68 


0039944 


29 


0016644 


59 


0033862 


29 


0019970 


59 


0040632 


30 


0017216 


60 0034434 


30 


0020660 


60 


0041322 




cc 



(218) 








Table for Estmating the Superficial Quantity 


of Slrnaa of 


Cuttings and Embankments in Rods, 




SLOPE i to 1. 


SLOPE i to 1. 


Ht 


CoBtenti 


Ht 


Contents 


Ht. 
1 


Contents 


Ht 


Contents 


A All* 


in Rodt. 


m* 


inRoda. 


in Rods. 


XI b. 


in Rods. 


1 


008786 


81 


117370 


004106 


81 


127305 


2 


007572 


32 


121166 


2 


008213 


32 


131412 


8 


011358 


33 


12494'i 


3 


012320 


38 


135519 


4 


016144 


34 


128728 


4 


016426 


34 


139626 


6 


018931 


85 


132515 


5 


020533 


35 


143732 


6 


022717 


36 


136301 


6 


024640 


36 


147839 


7 


026503 


37 


140087 


7 


028746 


37 


161945 


8 


030289 


38 


143873 


8 


032863 


38 


166052 


9 


084075 


39 


147689 


9 


036960 


39 


160159 


10 


087861 


40 


161445 


10 


041066 


40 


164266 


11 


041647 


41 


166231 


11 


045172 


41 


168372 


12 


045433 


42 


169017 


12 


049279 


42 


172479 


IS 


049219 


43 


162803 


13 


053386 


43 


176686 


14 


053005 


44 


166689 


14 


057492 


44 


180692 


15 


066792 


45 


170376 


16 


061699 


45 


184799 


16 


060578 


46 


174162 


16 


066706 


46 


188906 


17 


064364 


47 


177948 


17 


069812 


47 


193012 


18 


068160 


48 


181734 


18 


073919 


48 


197119 


19 


071936 


49 


186520 


19 


078026 


49 


201226 


20 


075723 


50 


189307 


20 


082133 


60 


206332 


21 


079509 


61 


193093 


21 


086239 


51 


209438 


22 


083295 


52 


196879 


22 


090346 


52 


213645 


23 


087081 


53 


200665 


23 


094453 


63 


217652 


24 


090867 


54 


204461 


24 


0986^ 


64 


221768 


25 


094654 


65 


208238 


26 


102666 


56 


225865 


26 


098440 


56 


212024 


26 


106773 


66 


229972 


27 


102226 


67 


215810 


27 


110879 


57 


234078 


28 


106012 


68 


219596 


28 


11^86 


58 


238185 


29 


109798 


59 


223382 


29 


119063 


59 


242292 


30 


113684 


60 227168 


30 


123199 


60 


246399 



(219) 

Tabk for Estimating the Superficial QHantity of Slopes of 
Cuttings and Embankments in Rods. 



Ht. 



1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

12 

13 

14 

16 

16 

17 

18 

19 

20 

21 

22 

23 

24 

25 

26 

27 

28 

29 

30 



SLOPE } to 1. 



I 



SLOPE 1 to 1. 



Contents 
in Rods. 



004591 
009183 
013774 
018365 
022957 
027548 
032139 
036731 
041322 
045914 
050505 
055097 
059688 
064279 
068871 
073462 
078053 
082645 
087236 
091827 
096418 
101010 
105601 
110192 
114784 
119375 
123966 
128558 
133149 
137741 



12^^ Contents 
in Rods. 



31 
32 
33 
34 
35 
36 
37 
38 
39 
40 
41 
42 
43 
44 
45 
46 
47 
48 
49 
50 
51 
52 
53 
54 
66 
66 
61 
58 
59 
60 



142332 
146924 
151515 
1561061 
160698 
165289 
169880 
174472 
179063 
183665 
188246 
192838 
197429 
202020 
206612 
211203 
215794 
220386 
224977 
229568 
234159 
238751 
243342 
247933 
252525 
257116 
261707 
266299 
270890 
275482 



Ht. 



1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

12 

13 

14 

15 

16 

17 

18 

19 

20 

21 

22 

23 

24 

25 

26 

27 

28 

29 

30 



Contents 
in Rods. 



005194 

010389 

015684 

020778 

025973 

031167 

036362 

041556 

046751 

051945 

057139 

062334 

067529 

072723 

077918 

083112 

088307 

093501 

098696 

103891 

109085 

114280 

119475 

124669 

129864 

135058 

140253 

145447 

150642 

155836; 



Ht. 



Contents 
in Rods. 



31 
32 
33 
34 
35 
36 
37 
38 
39 
40 
41 
42 
43 
44 
45 
46 
47 
48 
49 
50 
51 
52 
53 
54 
6^ 
61^ 
61 
58 
59 
60 
cc2 



161030 

166225 

171420 

176614 

181809 

187003 

192198 

197392 

202587 

207782 

212976 

218171 

223366 

228560 

233755 

238949 

244144 

249338 

254533 

259727 

264921 

270116 

275311 

280505 

285700 

290894 

296089 

301283 

306478 

311672 



(220) 

Table for Estimating the Superficial Quantity of Slopes of 
Cuttings and Embankments in Rods, 



SLOPE U to 1. 




SLOPE 1) to 


1. 


Ht 


Contents 


Ht. 


Contents 


Ht 


Contents 


Ht 


Contents 


XA !»• 


inRodi. 


AA V* 


in Rods. 


AA«» 


in Rods. 


JLXw* 


in Rods. 


1 


005880 


31 


182276 


1 


006^22 


31 


205275 


2 


011760 


32 


188165 


2 


013243 


32 


211896 


3 


017639 


33 


194034 


3 


919865 


33 


218518 


4 


023519 


34 


199914 


4 


026487 


34 


226140 


5 


029399 


35 


205794 


5 


033109 


36 


231762 


6 


035279 


36 


211674 


6 


039731 


36 


238384 


7 


041139 


37 


217634 


7 


046362 


37 


246005 


8 


047039 


38 


223434 


8 


052974 


38 


251627 


9 


052918 


39 


229313 


9 


059696 


39 


268249 


10 


058798 


40 


236193 


10 


066218 


40 


264871 


11 


064678 


41 


241073 


11 


072840 


41 


271493 


12 


070558 


42 


246963 


12 


079461 


42 


278114 


13 


076437 


43 


262832 


13 


086083 


43 


284736 


14 


082317 


44 


268712 


14 


092706 


44 


291368 


15 


088197 


46 


264592 


16 


099327 


45 


297980 


16 


094077 


46 


270472 


16 


106949 


46 


304602 


17 


099937 


47 


276332 


17 


112570 


47 


311223 


18 


105837 


48 


282232 


18 


119192 


48 


317845 


19 


111716 


^ 


287111 


19 


126814 


49 


324467 


20 


117596 


50 


293991 


20 


132435 


60 


331088 


21 


123476 


61 


299871 


21 


139057 


61 


337710 


22 


129356 


52 


306751 


22 


145678 


52 


344331 


23 


135235 


63 


311630 


23 


162300 


63 


360953 


24 


141115 


64 


317610 


24 


158922 


54 


367676 


26 


146995 


55 


323390 


26 


165544 


56 


364197 


26 


152875 


56 


329270 


26 


172166 


66 


370819 


27 


168735 


67 


336130 


27 


178787 


67 


377440 


28 


164635 


58 


341030 


28 


185409 


68 


384062 


29 


170515 


59 


346909 


29 


192031 


69 


390684 


30 


176395 


60 


352789 


30 


198653 


60 


397306 



(221) 


Table for Estimating the Svperfiaial Qaantil^ qf Slopet of 




SLOPE IJ to 1. 


SLOPE 2 to 1. j 


Ht 


CoDtents jjt 


Contents 




Contents 




Contents 




b Bods. "*■ 


inKotk. 


■ 


in Boas. 




in Bods. 


1 


007403 31 


229504 


1 


008213 


31 


254612 


2 


014807 32 


236908 


2 


016427 


32 


262825 


3 


022210 


33 


244311 


3 


024640 


33 


271039 


4 


029613 


34 


251714 


4 


032853 


34 


279252 


S 


037017 


35 


259118 


5 


041066 


35 


287465 


6 


044420 


36 


266521 


6 


049280 


36 


295679 


7 


051824 


37 


273925 


7 


057493 


37 


303892 


8 


059227 


38 


281328 


8 


065706 


38 


312105 


9 


066630 


39 


288731 


9 


073920 


39 


320319 


10 


074034 


40 


296134 


10 


082133 


40 


328532 


11 


081437 


41 


303637 


11 


090346 


41 


336745 


12 


088841 


42 


310941 


12 


098660 


42 


344959 


13 


096244 


43 


318344 


13 


106773 


43 


353172 


U 


103647 


44 


325747 


14 


114986 


44 


361385 


15 


111061 


45 


333151 


15 


123199 


45 


369598 


16 


118454 


46 


340654 


16 


131413 


46 


377812 


17 


125858 


47 


347968 


17 


139626 


47 


386025 


18 


133261 


48 


365361 


18 


147839 


48 


394238 


10 


140664 


49 


362764 


19 


156063 


49 


402462 


20 


148067 


50 


370168 


20 


164266 


50 


410664 


21 


156470 


51 


377571 


21 


172479 


51 


418877 


22 


162874 


52 


384975 


22 


180693 


62 


427091 


28 


170277 


53 


392378 


23 


188906 


63 


436304 


24 


177680 


54 


399781 


24 


197119 


64 


443517 


25 


185084 


55 


407183 


25 


205332 


55 


451730 


26 


192487 


56 


414588 


26 


213546 


66 


«99«l 


27 


199891 


57 


421992 


27 


221769 


67 


468ii»» 


28 


207294 


58 


429395 


28 


229972 


68 


29 


214697 


59 


436798 


29 


238186 


59 


48^^^V 


30 


222101 60 


444201 


30 


246399 


60 


_ji^ 



(222) 

Table for Estitruiting the Superficial Quantity of Slopes of 
Cuttings and Embankments in Rods, 



1 



Ht. 



1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

12 

13 

14 

15 

16 

17 

18 

19 

20 

21 

22 

23 

24 

25 

26 

27 

23 

29 

30 



SLOPB 2i to 1. 



Coatents 
in Rods. 



Ht. 



.ir4:ii 



019780 
029670 
039560 
049450 
0S9341 
069231 
079121 
089011 
098901 
108791 
118681 
128571 
138461 
148351 
158242 
168132 
178022 
187912 
197802 
207692 
217582 
227472 
237362 
247252 
257143 
267033 
276923 
286813 
296703 



31 
32 
33 
34 
35 
36 
37 
38 
39 
40 
41 
42 
43 
44 
45 
46 
47 
48 
49 
50 
51 
52 
53 
54 
55 
56 
67 
68 
69 
60 



Contents 
in Rods. 



306593 
316483 
326373 
336263 
346153 
356044 
365934 
375824 
385714 
395604 
405494 
415384 
425274 
435164 
445054 
454945 
464835 
474725 
484615 
4945a5 
504395 
514285 
524175 
634065 
543955 
553846 
563736 
573626 
583516 
593406 



SLOPE 3 to 1. 



Ht. 



1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

12 

13 

14 

15 

16 

17 

18 

19 

20 

21 

22 

23 

24 

25 

26 

27 

28 

29 

30 



Contents 
in Rods. 



011616 
023232 
034846 
046464 
058077 
069696 
081307 
092929 
104638 
116163 
127768 
139384 
151000 
162614 
174230 
185845 



197460 47 



Ht. 



31 
32 
33 
34 
35 
36 
37 
38 
39 
40 
41 
42 
43 
44 
46 
46 



Contents 
in Rods. 



209076 
2-20691 
232307 
243922 
255538 
267153 
278768 
290384 
292000 
313614 
325230 
336845 
348460 



48 
49 
60 
61 
52 
63 
64 
55 
66 
67 
58 
69 
60 



860076 
371691 
383306 
394921 
406537 
418162 
429767 
441383 
463000 
464614 
476229 
487845 
499460 
611076 
622691 
634306 
646921 
667637 
669162 
580767 
692382 
604000 
615613 
627228 
638844 
650469 
682074 
673690 
686305 
696921 



EXPLANATION OF TABLES 

Ftyr Calculating the Areas of Land required for a Railivay, 



Having prepared a form similar to that required for the 
earthwork, look in the tables for the given slope, opposite 
the given heights at each end of the prismoid, in the 
columns headed " sides ;" and also look in the columns 
headed ''bases/' which corresponds with slope 1 to 1 of the 
sides, for a number corresponding with the given base added 
to the two widths on each side between the edge of the 
slope at the surface of the ground, and the boundary 
or fence line. These three numbers added together and 
multiplied by the length will give the area in acres and 
decimals : all dimensions are given in feet. 

Example : 

Slope 1 to 1 ; base 32 feet. 

Side widths, 12 feet each. 

Heights, 18 and 38 feet; length, 560 feet. 

Col. 1 Opposite 18 we find - - .0004132 

Sides./Opposite38wefind - - .0008723 

bScs \ Opposite 32 + 12 + 12 = 56 we find .0012855 



.0025710 
Multiply by length = 560 

1.4397600 

A. R. P. 

Answer 1 1 30 



224 

It is obvious that in any given railway, whose base 
and side widths are determined, the third number (in this 
case .0012855) is common to all the additions ; therefore^ 
this may be multiplied by the entire length at once, and 
the sides computed and added to the result afterwards. 
Moreover, the same as in the earthwork, one tabular 
number is common to two adjoining plots. Much time 
may be saved also, by adding mentally the two heights 
together, and seeking, if the tables admit of it, the corres- 
ponding tabular number ; for 

Opposite 18 we find .0004132 

Opposite 38 " .0008723 

Their sum is 56 .0012855 

and opposite 56 we find .0012855. 

Table to find the Superficial Quantity of Slopes. 

Add together the numbers opposite the given heights, 
and multiply the sum by the length ; the product in the 
area is rods and decimals. 

Example : the heights and slopes as before : 
Opposite 18 we find .093501 
Opposite 38 " .197392 

56 .290893 

Multiply by length 560 

RODS. YDS. 

Answer 162 27 162.900080 

Here again, the sum of the heights being taken = 66, 
the corresponding number is .290893. 



- — " 









( 225 ) 




TABLE OF OFFSETS TO CURVES, 




Calculated for Radius = 1000. 


Tan. 


Offsett. 


Tan. 


Offsett. 


Taa. 


CMbett 


Tan. 


Offiwtt. 


1 


.0006 


31 


.4806 


61 


1.8623 


91 


4.1492 


2 


.0020 


32 


.5121 


62 


1.9239 


92 


4.2411 


3 


.0045 


33 


.5446 


63 


1.9865 


93 


4.3340 


4 


.0080 


34 


.5781 


64 


2.0501 


94 


4.4279 


5 


.0125 


35 


.6127 


65 


2.1148 


95 


4.6228 


6 


.0180 


36 


.6482 


66 


2.1804 


96 


4.6187 


7 


.0245 


37 


.6847 


67 


2.2471 


97 


4.7157 


8 


.0320 


38 


.7222 


68 


2.3147 


98 


4.8136 


9 


.0406 


39 


.7607 


69 


2.3833 


99 


4.9216 


10 


.0501 


40 


.8003 


70 


2.4530 


100 


5.0126 


11 


.0606 


41 


.8408 


71 


2.6237 


101 


5.1136 


12 


.0721 


42 


.8824 


72 


2.6954 


102 


5.2167 


13 


.0846 


43 


.9249 


73 


2.6681 


103 


5.3187 


14 


.0981 


44 


.9685 


74 


2.7418 


104 


6.4228 


15 


.1126 


45 


1.0131 


75 


2.8165 


105 


6.5278 


16 


.1281 


46 


1.0586 


76 


2.8922 


106 


5.6339 


17 


.1446 


47 


1.1052 


77 


2.9689 


107 


5.7411 


18 


.1621 


48 


1.1528 


78 


3.0466 


108 


6.8492 


19 


.1806 


49 


1.2013 


79 


3.1253 


109 


5.9584 


20 


.2001 


50 


1.2508 


80 


3.2052 


110 


6.0686 


21 


.2206 


51 


1.3013 


81 


3.2859 


111 


6.1796 


22 


.2421 


52 


1.3529 


82 


3.3677 


112 


6.2918 


23 


.2646 


53 


1.4054 


83 


3.4506 


113 


6.4061 


24 


.2881 


54 


1.4590 


84 


3.6343 


114 


6.5193 


25 


.3126 


55 


1.5137 


85 


3.6191 


115 


6.6346 


26 


.3381 


56 


1.5693 


86 


3.7049 


116 


6.7608 


27 


.3646 


57 


1.6259 


87 


3.7917 


117 


6.8681 


28 


.3921 


58 


1.6835 


88 


3.8795 


118 


6.9865 


29 


.4206 


59 


1.7421 


89 


3.9684 


119 


7.1068 


30 


.4501 


60 


L8017 


90 


4.0638 


120 


7.2262 



:l.v 



(226) 


TABLE OF OFFSETS TO CURVES, 


Calculated for Radius = 1000. 


Tin. 


OfEKtt. 


Tan. 


Offtett. 


Tan. 


Offfett. 


Tan. 


Offiwtt. 


121 


7.3474 


151 


11.4661 


181 


16.5168 


211 


22.5139 


122 


7.4797 


152 


11.6193 


182 


16.7013 


212 


22.7303 


123 


7.5931 


153 


11.7786 


183 


16.8869 


213 


22.9479 


124 


7.7176 


154 


11,9290 


184 


17.0736 


214 


23.1665 


125 


7.8433 


156 


12.0856 


185 


17.2615 


215 


23.3860 


126 


7.9696 


166 


12.2428 


186 


17.4502 


216 


23.6066 


127 


8.0971 


167 


12.4013 


187 


17.6400 


217 


23.8283 


128 


8.2276 


168 


12.5609 


188 


17.8310 


218 


24.0511 


129 


8.3572 


160 


12.7213 


189 


18.0229 


219 


24.2761 


130 


8.4861 


160 


12.8830 


190 


18.2160 


220 


24.5002 


131 


8.6175 


161 


13.0455 


191 


18.4099 


221 


24.7262 


132 


8.7501 


162 


13.2091 


192 


18.6050 


222 


24.9583 


133 


8.8837 


163 


13.3739 


193 


18.8012 


223 


25.1815 


134 


9.0185 


164 


18.5396 


194 


18.9986 


224 


25.4108 


135 


9.1544 


165 


13.7065 


195 


19.1968 


225 


25.6413 


136 


9.2910 


166 


13.8742 


196 


19.3960 


226 


25.8727 


137 


9.^87 


167 


14.0480 


197 


19.5964 


227 


26.1052 


138 


9.5675 


168 


14.2130 


198 


19.7978 


228 


26.3388 


139 


9.7074 


169 


14.3839 


199 


20.0004 


229 


2a5735 


140 


9.8485 


170 


14.5560 


200 


20.2041 


230 


26.8094 


141 


9.9903 


171 


14.7290 


201 


20.4087 


231 


27.0463 


142 


10.1332 


172 


14.9080 


202 


20.6144 


232 


27.2842 


143 


10.2772 


173 


15.0782 


203 


20.8212 


233 


27.5233 


144 


10.^23 


174 


16.2545 


204 


21.0291 


234 


27.7634 


145 


10.5685 


175 


15.4318 


205 


21.2381 


235 


28.0047 


146 


10.7154 


176 


15.6099 


206 


21.4480 


236 


28.2470 


147 


10.8684 


177 


15.7891 


207 


21.6590 


237 


28.4904 


148 


110126 


178 


16.9694 


206 


21.8711 


238 


28.7Md 


149 


11.1687 


179 


16.1508 


209 


22.0843 


239 


28.9805 


150 

- 


11.3140 


180 


16.3334 


210 


22.2987 


240 29.2271 



(227) 
TABLE OF OFFSETS TO CURVES, 

Calculated for Radius = 1000. 



Tan. 



241 

242 
243 
244 
245 
246 
247 
248 
249 
250 
251 
252 
253 
254 
266 
256 
267 
258 
259 
260 
261 
262 
263 
264 
265 
266 
267 
268 
260 
270 



Offsett. 



29.4746 
29.7234 
29.9734 
30.2246 
30.4770 
30.7300 
30.9842 
31.2396 
31.4963 
31.7542 
32.0126 
32.2722 



271 
272 
273 

274 
275 
276 
277 
278 
279 
280 
281 
282 



32.5331 283 



: 



32,7952 
33.0686 
33.3226 
33.6880 287 
33.8546 288 
34.1225 289 
34.3914 290 
34.6610 291 
34.9319 292 
35.2039 293 
35.4772 294 
35.7516 295 
36.0268 296 
36.3032 297 
36.6809 298 
36.8597 299 
37.1397 300 



Tan. 



284 
286 
286 



Offsett. 






37.4206 

37.7026 

37.9867 

38.2701 

38.5568 

38.8422 

39.1298 

39.4186 

39.7086 

40.0000 

40.2921 

40.5864 

40.8799 

41.1766 

41.4726 

41. 

42,0692 

42.3694 

42.6708 

42.9734 

43.2768 

43.5815 

43.8874 

44.1946 

44.5028 

44.8120 

45.1224 

46.4340 

46.7468 

46.0608 



Tan. 



301 
302 

303 
304 
306 
306 
307 
308 
309 
310 
311 
312 
313 
314 
315 
7702 316 
317 
318 
319 
320 
321 
322 
323 
824 
325 
326 
327 
328 
329 
330 



Offsett. 



46.3757 
46.6919 
47.0093 
47.3279 
47.6477 
47.9684 
48.2903 
48.6135 
48.9379 
49.2635 
49.5900 
49.9178 
50.2468 
60.5770 
60.9084 
51.2408 
51.5744 
51.9092 
52.2453 
52.5825 
52.9208 
53.2603 
63.6010 
53.9429 
64.2860 
54.6302 
54.9756 
56.3223 
55,6701 
56.0191 



Tan. 



331 

332 
333 
334 
335 
336 
337 
338 
339 
340 
341 
342 
343 
344 
346 
346 
347 
348 
349 
350 
351 
352 
353 
354 
356 
356 
367 
368 
368 
360 



Offsett. 



56.3692 
56.7206 
57.0732 
57.4270 
57.7819 
68.1380 
58.4954 
58.8539 
59.2137 
59.5746 
59.9367 
60.3001 
60.6646 
61.0304 
61.3973 
61.7656 
^.1349 
62,6056 
62.8773 
63.2603 
63.6246 
64.0000 
64.3767 
64.7546 
66.1337 
66.5141 
66.8957 
66.2785 
66.6625 
67.0477 



i 



EXPLANATION 

OF THE 

TABLES OF OFFSETS TO CURVES. 



In setting out the curves of a railway in the fields, the 
usual practice is to lay out a tangent Ime, and from cer- 
tain points along this line to set off at right angles such 
distances as will give points in the curve : each distance is 
called "an offset/* It has generally been considered suflBi- 
cient, in setting out these curves, to calculate the offsets for 
every chain of tangent, to the extreme length of 10 chains; 
and the ordinary rule for this purpose (which is sufficiently 
near when the radius is not less than two miles) is to com- 
pute the extreme or last offset correctly^ and then all the 
others are derived from it, by making the offsets vary as the 
square of the tangent. But as the radius must be deter* 
mined for every set, and therefore the application of a 
table so constructed would be limited to a few given cases 
only, I thought it better to provide such a table as would 
be applicable to all cases, whether the radius were long or 
short ; and the tangent might bear such a proportion to 
the radius as would admit of the tables being used in 
very sharp curves. For instance, in a curve of 10 chains 
radius^ the tables wiU extend to 3.6 chains of tangent. 




/ 



/ 



239 

I now proceed to shew how these tables were formed, 
I i 6 I S ^'"^ ^^^^ ^° explain their use. 

Let AcC (in the accompanying 
diagram) be the arc of a circle 
required to be laid down ; AD is 
cthe radius, AB the tangent, and 
BC the offset. Abo Ac £c are 
intermediate offsets. AC is the 
chord. By the well known equa- 
tion of a circle we hav e this rule to 
findBC. BC=AI>— VaD* — AB'. 
Suppose that AD were 1000, and 
that tangents were given in the 
ZJ series 1, 3, 8, 4, &c., corresponding 

to A J, Ai, &c., then the offsets .0005, .0030, .0045, .0080, 
&c., are the true distances be be, kc. 

In the tables, every consecutive fifth number was calcu- 
lated according to the above rule ; and the intermediate 
numbers were interpolated. The error {if any) is m the 
4th place of decimals only, and then not exceedmg any 
where .0003; and this discrepancy, with a radius of 1000 
chains, or 12j miles, will not equal j of an inch. There- 
fore, the tables may be considered as practically accurate. 

KDLB FOR THE TABLES. 

Given radina and tangent, to find the offset. Divide 
1 000 times the tangent by radius, and look in column of 
t^gents for the quotient; then multiply the correspond, 
ing opposite offset by 1,000 times the radius : the result is 
the true offset. 



= 50. 



Radius, 3,940; tangent, 147. Here ^^^3^0°" 

In the column of tangents in the tables we lot 
and opposite to this, in the column of offset 
1 2508 : this number multiplied by the rad 
aiid divided by 1000, wiU pve the true offs 

_ 1.2508 "2940 _ 8_677, 
"" 1000 



280 

It is very easy to assume such a tangent as will give^ 
when divided, some nimiber to be found in the table; 
which number should again admit of being sub-divided 
into equal parts by some convenient divisor. For instance, 
the above example (147) is easily divided by 2,940, and the 
quotient 60 is found in the table. Also 50 can be divided 
easily by 5 or 10, and will admit of 10 offsets 14.7 apart, 
or 5 offsets 29.4 apart. In the first case, the numbers 
opposite 5, 10, 15, &c., must be chosen to find the offsets; 
and in the latter case, we should extract the numbers 
opposite 10, 20, 30, &c. ; and computing these offsets as 
b^ore, we can fill up so many points in the curve. 

The use of these tables affords equal facility, whether 
the calculations are made in feet, chains, yards, or any 
other measure. Besides the use of computing the offsets 
to curves, for small divisions on a tangent line, it is some- 
times very desirable to know some remote point in a long 
curve, for the purpose of checking the work, or when the 
instruments for delineating curves are not long enough 
to reach the whole extent of the required sweep. For 
instance, in a working plan, plotted to 2^ chains to 1 inch, 
or 32 inches to 1 mile, suppose a curve of 4 miles' radius 
to extend over 2 miles of Ime, this will be 5 feet 4 inches 
long ; and as the radius is 10 feet 8 inches, unless some 
points were computed, it would not be safe to depend upon 
the line which could be drawn by a sweep only 18 or 24 
inches long. But, to make some of the points through which 
the curve should pass, calculate them by the tables; thus: 

Given radius 4 miles = 320 chains; tangents, 16, 

32, 48, 64, 80 chains ; here, ?OjLl22? = 250 ; and as 250 
divides easily by 5, we can extract the numbers opposite. 

CHAINS. 

250 = 31.7542^ then each f Offset for tangent 80 

200 == 20.2041 number j '' '' 64 

150 = 11.3140^ multiplied <{ '' '' 48 

100 = 5.0126 1 by I '' '' 32 

50 = 1.2508J 320-rl000l " " 16 



231 

It is now easy (having plotted these offsets) to join the 
points by a proper sweep. To facilitate various calcula- 
tions relating to curves, I subjoin the following rules : — 

1. — Given tangent AB and offset BC ; to find radius 
AD = AD : 

AB^ + BC? Add together the squares of the tangent 
2 BC and offset, and divide by twice the offset ; the 
quotient is the radius. 

2. — Given chord AC, and offset BC ; to find radius 
AD = AD : 

AC^ Divide the square of the chord by twice the 

2 BC offset ; the quotient is the radius. 

BC == 3. — Given radius and tangent ; to find the offset : 

From the radius deduct the square root of 
A D — the difference of the squares of the radius and 

VAD^ AW tangent ; the result is the offset. 

BC = 4. — Given the radius and chord ; to find the offset : 
AC^ Divide the square of the chord by twice the 

2 AD radius ; the quotient is the offset. 

AB = 5. — Given the chord and offset ; to find the tangent. 

VaC^— BC^ Take the square root of the difference of the 
squares of the chord and offset ; the result is 
the tangent. 

AB = 6. — Given the radius and offset; to find the tangent: 

V2 AD X BC Multiply the difference of twice the radius 
— BC" and the offset by the offset ; and the square 

root of the product is the tangent. 
7. — Given the tangent and offset; to find the 
AC = chord: 

VaB^Tbc? Add together the squares of the tangent 

and offset ; the square root of the sum is the 

chord. 
8. — Given the radius and offset; to find the 
AC = chord : 

V2 BC X AD Multiply twice the offset by the radius j the 

square root of the product is the chord. 



1 



232 

TMe for finding the Radius, when the Tangent is knoum, 
and the AngU subtending the are is given (contained 
between the two tangents to the extreme points of the arc). 

'Rxiij&—To find the radius. 
Multiply the tangent by the coefficient opposite the 

given angle; the product is the radius. 



Angle. 
179i 
179 
1784 
178 
1774 
177 
176 J 
176 
175| 
175 
174* 
174 
173i 
173 
172i 
172 
171J 
171 
170i 
170 
169i 
169 
168} 
168 
167* 
167 
166* 
166 
165* 
165 



Coefficient. 
229.18 
114.60 
76.40 
57.30 
45.84 
38.20 
32.74 
28.65 
25.47 
22.92 
20.84 
19.10 
17.64 
16.38 
15.29 
14.33 
13.49 
12.74 
12.07 
11.47 
10.92 
10.43 
9.981 
9.567 
91.85 
8.833 
8.507 
8.205 
7.924 
7.661 



II Angle. 


Coefficient 


164* 


7.415 


164 


7.185 


163j 


6.969 


163 


6.765 


162i 


6.573 


162 


6.392 


161* 


6.221 


161 


6.059 


leoj 


5.905 


160 


5.758 


159* 


5.619 


159 


5.487 


158* 


5.361 


158 


5.241 


157* 


5.125 


157 


5.015 


156i 


4.910 


156 


4.809 


155i 


4.713 


155 


4.620 


154i 


4.531 


154 


4.445 


153i 


4^363 


153 


4.283 


152i 


4.207 


152 


4.133 


151* 


4.062 


151 


3.994 


150j 


3.927 


150 


3.863 



233 

Given radius and angle ; to find corresponding tangent. 

Rule — ^Divide radius by coefficient due to the angle; 
the quotient is the required tangent. 



EXPLANATION OF THE TABLE OF DISTANCES. 

For the convenience of those who wish to transfer chains 
into feet or yards, or the reverse, I have supplied tables, 
giving the comparative values of any desired distances at 
once ; and by multiplying or dividing the numbers given 
in the table by 10, it is easy to obtain any extension of the 
tables required in practice. 



B D 



A 



(234) 

TABLE OF DISTANCES. 

Chains into Yards and Feet. 



Miles 


Chains. 


YariU. 


Feet. 


Miles 


Chains. 


Yards. 


Feet. 




.01 


.22 


.66 


11 


242 


726 




.02 


.44 


1.32 


i 12 


264 


792 




.03 


.66 


1.98 




13 


286 


858 




.04 


.88 


2.64 




14 


306 


924 




.05 


1.10 


3.30 


,'< 


16 


330 


990 




.06 


1.32 


3.96 




16 


352 


1056 




.07 


1.54 


4.62 




17 


374 


1122 




.08 


1.76 


5.28 




18 


396 


1188 




.09 


1.98 


5.94 




19 


418 


1254 




.10 


2.20 


6.60 


i 


20 


440 


1320 




.20 


4.40 


13.20 




21 


462 


1386 




.30 


6.60 


19.80 




22 


484 


1452 




.40 


8.80 


26.40 




23 


506 


1518 




.50 


11.00 


33.00 




24 


528 


1584 




.60 


13.20 


39.60 


6 
16 


25 


550 


1650 




.70 


15.40 


46.20 




26 


572 


1716 




.80 


17.60 


52.80 




27 


594 


1782 




.90 


19.80 


59.40 




28 


616 


1848 




1.00 


22.00 


66.00 




29 


638 


1914 




2.00 


44.00 


132.00 


1 


30 


660 


1980 




3.00 


66.00 


198.00 




31 


682 


2046 




4.00 


88.00 


264.00 




32 


704 


2112 




5.00 


110.00 


330.00 




33 


726 


2178 


1 

u 


6.00 


132.00 


396.00 




34 


748 


2244 




7.00 


154.00 


462.00 


7 
16 


35 


770 


2310 




8.00 


176.00 


528.00 




36 


792 


2376 




9.00 


198.00 


594.00 




37 


814 


2442 


i 


10.00 


220.00 


660.00 




38 
39 


836 

858 


2508 
2574 










h 


40 


880 


2640 



I 

I 







1 

(235) 


TABLE 


. OF DISTANCES. 1 




Chains into Yards and Feet. 

— — — >^ 1 


Miles 




Yard*. 


Feet. 


Mileg 


Clwiiii. 


Yard*. 


Feet 




41 


902 


2704 




71 


1662 


4686 




42 


924 


2772 


1 72 


1584 


4762 




43 


946 


2838 


1 73 


1606 


4818 




44 


968 


2904 


74 


1628 


4884 


9 
16 


45 


990 


2970 


ij ; 76 


1660 


4950 




46 


1012 


3036 


76 


1672 


6016 




47 


1034 


3102 


, 77 


1694 


6082 




48 


1056 


3168 


i 78 


1716 


6148 




49 


1078 


3234 


79 


1738 


5214 


1 


50 


1100 


3300 


1 80 


1760 


5280 




51 


1122 


3366 


81 


1782 


5346 




52 


1144 


3432 


i 82 


1804 


5412 




53 


1166 


3498 


; 83 


1826 


6478 




54 


1188 


3564 


84 


1848 


6544 


11 
16 


55 


1210 


3630 


1.', 85 


1870 


5610 




56 


1232 


3696 




86 


1892 


5676 




57 


1254 


3762 


1 87 


1914 


6742 




58 


1276 


3828 




88 


1936 


5808 




59 


1298 


3894 




89 


1958 


6874 


1 


60 


1320 


3960 


U 


90 


1980 


5940 




61 


1342 


4026 




91 


2002 


6006 




62 


1364 


4092 




92 


2024 


6072 




63 


1386 


4158 




93 


2046 


6138 




64 


1408 


4224 




94 


2068 


6204 


13 
16 


65 


1430 


4290 


1?, 


95 


2090 


6270 




66 


1462 


4356 




96 


2112 


6336 




67 


1474 


4422 




97 


2134 


6402 




68 


.1496 


4488 




98 


2156 


6468 




69 


1518 


4554 




99 


2178 


6534 


i 


70 


1540 


4620 


u 


100 


2200 


6600 






D d2 



j 



I 

J 







(£36) 






TABLE 


OF DISTANCES. 




Chains into Yards and Feet. 


Mile* 


Chains. 


Yards. 


7«et. 


Miles 

41 


Chains. 


Yards. 


Jrcct* 


1.'. 


105 


2310 


6930 


350 


7700 


23100 


li 


110 


2420 


7260 


4i 


360 


7920 


23760 


17. 


115 


2630 


7590 


4* 


370 


8140 


24420 


li 


120 


2640 


7920 


4i 


380 


8360 


25080 


U 


125 


2750 


8260 


45 


390 


8680 


25740 


1* 


130 


2860 


8680 


5 


400 


8800 


26400 


US 


ia5 


2970 


8910 


6J 


410 


9020 


27060 


11 


140 


3080 


9240 


6i 


420 


9240 


27720 


Hi 


146 


3190 


9570 


51 


430 


9460 


28380 


IJ 


1-50 


3300 


9900 


5i 


440 


9680 


29040 


115 

•••18 


156 


3410 


10230 


51 


450 


9900 


29700 


2 


160 


3620 


10560 


4i 


460 


10120 


30360 


2i 


170 


3740 


11220 


51 


470 


10340 


31020 


2i 


180 


3960 


11880 


6 


480 


10560 


31680 


21 


190 


4180 


12540 


6t 


490 


10780 


32340 


2i 


200 


4400 


13200 


6i 


600 


11000 


33000 


2f 


210 


4620 


13860 


61 


610 


11220' 


33660 


2i 


220 


4840 


14520 


6i 


520 


11440 


34320 


2i 


230 


5060 


15180 


61 


630 


11660 


34980 


3 


240 


5280 


16840 


6i 


540 


11880 


a5640 


3i 


250 


5600 


16500 


6J 


560 


12100 


36300 


3i 


260 


5720 


17160 


7 


560 


12320 


36960 


31 


270 


6940 


(7820 


7J 


570 


12540 


37620 


3i 


280 


6160 


18480 


7J 


580 


12760 


38280 


31 


290 


6380 


19140 


71 


690 


12980 


38940 


3i 


300 


6600 


19800 


7i 


600 


13200 


39600 


3J 


310 


6820 


20460 


7f 


610 


13420 


40260 


4 


320 


7040 


21120 


71 


620 


13640 


40920 


4J 


330 


7260 


21780 


75 


630 


13860 


41580 


4i 


340 


7480 


22440 


8 


640 


14080 


42240 



' 237 ■ 

TABLE OF DISTANCES. 

Yards inia Cbams aad Feet. 



Feet. 


Ta^. 


o^ 


Fort. 


Tardk 


1**^ 


.1 


.0^ 


-CBllo * 


21 


, 7-000 


.3181 


51 


.006 


/-■>30 


-22 


, 7.333 


.3333 


^ 


.100 


■ .Ci>4o 


23 


7.666 


.3184 


A 


.133 


.^JiJdiJ 


21 


8.000 


.3636 


.5 


.166 


-tJjTo 


25 


8.3^ 


.3787 


.6 


J200 


-OiJOO 


26 


8.666 


.3939 


.7 


.233 


.0106 


27 


9.000 


.4090 


^ 


J266 


-0121 • 


28 


9.333 


.ASASt 


.9 


.300 ' 


.0136 i 


29 


9.666 


.4393 


1.0 


.333 


.0151 ! 


30 


lO.OOO 


.4545 


2.0 


.oob 


.0303 


31 


10.333 


.4696 


3.0 


1.000 


.0154 


32 


10.666 


.4848 


4.0 


1.333 


.0606 


33 


11.000 


.5000 


5.0 


1.666 


.0757 


34 


11.333 


.5151 


6.0 


2.000 


.0909 


35 


11.666 


.5303 


7.0 


2.333 


.1060 


36 


12.000 


.5454 


8.0 


2.666 


.1212 


37 


12.333 


.5606 


9.0 


3.000 


.1363 


38 


12.666 


.5757 


10.0 


3.333 


.1515 


39 


13.000 


.5909 


11.0 


3.666 


.1666 


40 


13.333 


.6060 


12.0 


4.000 


.1818 


41 


13.666 


.6212 


13.0 


4.333 


.1969 


42 


14000 


.6363 


14.0 


4.666 


.2121 


43 


14.333 


.6515 


15.0 


5.000 


.2272 


44 


14.666 


.6666 


16.0 


5.333 


.2421 


45 


15.000 


.6818 


17.0 


5.666 


.2575 


46 


15.333 


.6969 


18.0 


6.000 


.2727 


47 


15.666 


.7121 


19.0 


6.330 


.2878 


48 


16.000 


.7272 


20.0 


6.666 


.3030 


49 


16.333 


.7424 








50 


16.C6(i 


.7676 



11 



|l 







(238) 








TABLE OF DISTANCES. 




Feet into Chaing and Yards. 




Yards. 


Feet. 


Chung. 


Yards. 


Feet* 


Chains. 


.1 


.8 


.0045 


21 


63 


.9545 


.2 


.6 


.0090 


22 


66 


1.0000 


.8 


.9 


.0136 


23 


69 


1.0464 


.4 


1.2 


.0181 


24 


72 


1.0909 


.6 


1.6 


.0227 


26 


75 


1.1363 


.6 


1.8 


.0272 


26 


78 


1.1818 


.7 


2.1 


.0318 


27 


81 


1.2272 


.8 


2.4 


.0363 


28 


84 


1.2727 


.9 


2.7 


.0409 


29 


87 


1.3181 


1.0 


8.0 


.0454 


30 


90 


1.3636 


2.0 


6.0 


.0909 


81 


98 


1.4090 


3.0 


9.0 


.1363 


32 


96 


1.4545 


4.0 


12.0 


.1818 


33 


99 


1.5000 


5.0 


15.0 


.2272 


34 


102 


1.6464 


6.0 


18.0 


.2727 


35 


105 


1.5909 


7.0 


21.0 


.3181 


36 


108 


1.6363 


8,0 


24.0 


.3636 


37 


111 


1.6818 


9.0 


27.0 


.4090 


38 


114 


1.7272 


10.0 


30.0 


.4546 


39 


117 


1.7727 


11.0 


33.0 


.5000 


40 


120 


1.8181 


12.0 


36.0 


.6454 


41 


128 


1.8636 


13.0 


39.0 


.5909 


42 


126 


1.9090 


14.0 


42.0 


.6363 


43 


129 


1.9645 


15.0 


45.0 


.6818 


44 


132 


2.0000 


16.0 


48.0 


.7272 


45 


135 


2.0464 


17.0 


51.0 


.7727 


46 


138 


2.0909 


18.0 


64.0 


.8181 


47 


141 


2.1363 


19.0 


57.0 


.8636 


48 


144 


2.1818 


20.0 


60.0 


.9090 


49 


147 


2.2272 








60 


150 


2.2727 





(289) 


1 
1 




TABLE OP GRADIENTS. 


1 




•S-S 


Vert, alt 


. Vert, alt 


. Gravity in 


cTS 


Vert alt. 


Vert. alt. 


Gnu in 






4 S 


in feet 


in feet 


lb«. due 


•|» 


in feet 


in feet 


lbs. dne 






e3§ 
22 


per mile 


, perchaii 


1 to I ton. 


i2§ 

52 


per mile. 


perchaii 


to 1 ton. 






240.0C 


1 3.000C 


1101.816 


101.53 


1.2692 


43.076 






23 


229.56 


2.8696 


97.368 


53 


99.62 


1.2462 


42.264 






24 


220.00 


2.7500 


93.336 


54 


97.77 


1.2222 


41.480 






25 


211-20 


2.6400 


89.600 


55 


96.00 


1.2000 


40.726 






26 


203.06 


2.5384 


86.152 


56 


94.28 


1.1786 


40.000 






27 


195-55 


2.4444 


82.960 


57 


92.63 


1.1578 


39.298 






28 


188.56 


2.3572 


80.000 


58 


91.03 


1.1377 


38.620 






29 


182.06 


2.2758 


77.240 


59 


89.49 


1.1186 


37.966 






30 


176.00 


2.2000 


74.666 


60 


88.00 


1.1000 


37.333 






31 


170.32 


2.1290 


72.216 


61 


86.55 


1.0818 


36.720 






32 


165.00 


2.0625 


70.000 


62 


85.16 


1.0645 


36.108 






33 


160.00 


2.0000 


67.880 


63 


83.81 


1.0477 


35.555 






34 


155.30 


1.9413 


65.880 


64 


82.50 


1.0312 


35.000 






35 


150.84 


1.8856 


64.000 


65 


81.23 


1.0152 


34.460 






36 


146.66 


1.8333 


62.222 


66 


80.00 


1.0000 


33.940 






37 


142.70 


1.7839 


60.540 


67 


78.81 


.9851 


33.432 






38 


138.95 


1.7368 


58.944 


68 


77.64 


.9706 


32.940 






39 


135.38 


1.6923 


57.436 


69 


76.49 


.9563 


32.464 






40 


132.00 


1.6500 


56.000 


70 


75.43 


.9429 


32.000 






41 


128.78 


1.6098 


54.634 


71 


74.36 


.9296 


31.660 






42 


125.71 


1.5714 


53.333 


72 


73.33 


.9166 


31.111 






43 


122.78 


1.5348 


52.092 


73 


72.32 


.9041 


30.685 






44 


120.00 


1.5000 


50.908 


74 


71.35 


.8919 


30.270 






45 


117.33 


1.4666 


49.777 


75 


70.40 


.8800 


29.867 






46 


115.04 


1.4380 


48.684 


76 


69.47 


.8684 


29.472 


4 




47 


112.34 


1.4042 


47.660 


77 


68.57 


.8570 


29.090 




■ 


48 


110.00 


1.3750 


46.688 


78 


67.69 


.8461 


28.718 






49 


107.75 


1.3470 


45.716 


79 


66.83 


.8355 


28.355 






60 


105.60 


1.3200 


44.800 


80 


66.00 


.8250 


28.000 






51 103.52 


1.2941 


43.920 


81 


65.20 1 


.8140 


27.718 





(240) 
TABLE OF GRADIENTS. 



Ratio. 


Ver.tl. 


Ver.tl. 


Ora. in 


T% A • 


Ver. al 


Ver.al. 


Gra. in 


in feet 


in feet 


lbs. due 


Ratio. 


in feet 


in feet 


Ibi. due 


One in 


per m. 


per ch. 


to 1 ton. 


One in 


per m. 


per ch. 


to I ton. 


82 


64.39 


.8048 


27.317 


no 


48.00 


.6000 


20.363 


83 


63.61 


.7952 


26.988 


111 


47.57 


.5940 


20.180 


84 


62.86 


.7857 


26.666 


112 


47.14 


.5893 


20.000 


85 


62.12 


.7764 


26.353 


113 


46.72 


.5840 


19.823 


85.16 


62.00 


.7750 


26.303 


114 


46.31 


.6789 


19.649 


86 


61.39 


.7674 


26.046 


115 


45.93 


.5739 


19.478 


87 


60.69 


.7586 


25.746 


116 


45.52 


.5688 


19.310 


88 


60.00 


.7500 


25.454 


117 


45.13 


.5641 


19.145 


89 


59.33 


.7416 


25.168 


118 


44.75 


.5593 


18.983 


90 


68.66 


.7333 


24.888 


119 


44.37 


.5545 


18.820 


91 


58.02 


.7252 


24.614 


120 


44.00 


.5500 


18.666 


92 


57.62 


.7190 


24.342 


121 


43.64 


.5464 


18.512 


93 


56.77 


.7096 


24.086 


122 


43.28 


.5409 


18.360 


94 


56.17 


.7021 


23.830 


123 


42.92 


.5364 


18.210 


95 


55.60 


.6900 


23.679 


124 


42.58 


.5322 


18.054 


96 


55.00 


.6875 


23.334 


125 


42.24 


.5280 


17.920 


97 


54.43 


.6804 


23.092 


126 


41.90 


.6238 


17.777 


98 


53.88 


.6735 


22.858 


127 


41.57 


.5196 


17.638 


99 


53.33 


.6666 


22.626 


128 


41.26 


.5157 


17.500 


100 


52.80 


.6600 


22.400 


128.78 


41.00 


.5126 


17.394 


101 


52.27 


.6535 


22.180 


129 


40.93 


.6016 


17.364 


102 


51.76 


.6470 


21.960 


130 


40.61 


.5076 


17.230 


103 


61.26 


.6408 


21.747 


131 


40.08 


.5009 


17.099 


104 


50.77 


.6346 


21.538 


132 


40.00 


.6000 


16.970 


105 


50.28 


.6285 


21.333 


133 


39.70 


.4962 


16.842 


105.6 


50.00 


.6250 


21.212 


134 


39.40 


.4926 


16.716 


106 


49.81 


.6226 


21.132 


135 


39.11 


.4888 


16.692 


107 


49.34 


.6168 


20.934 


136 


38.82 


.4853 


16.470 


108 


48.89 


.6111 


20.740 


137 


38.64 


.4816 


16.360 


109 


48.44 


.6056 


20.655 


138 


38.24 


.4782 


16.232 



I- 



(241) 
TABLE OF GRADIENTS. 



Ratio. 


Ver. al. 


Ver. al. 


Ora. in 


^r« M.9 


Ver.al. 


Ver.al. 


Gnu in 


in feet 


in feet 


lbs. due 


Ratio. 


in feet 


in feet 


lbs. doe 


One in 


per m. 


per ch. 


to I ton. 


One in 


per m. 


per ch. 


to 1 ton. 


139 


38.00 


.4750 


16.114 


168 ' 


31.43 


.3929 


13.333 


140 


37.71 


.4713 


16.000 


169 


31.24 


.3904 


13.254 


141 


37.44 


.4680 


15.886 


170 


31.06 


.3883 


13.176 


142 


37.18 


.4648 


15.775 


171 


30.88 


.3869 


13.099 


143 


36.92 


.4614 


15.666 


172 


30.69 


.3837 


13.023 


144 


36.66 


.4583 


15.555 


173 


30.57 


.3821 


12.942 


145 


36.41 


.4551 


15.448 


174 


30.34 


.3793 


12.873 


146 


36.16 


.4520 


15.342 


175 


30.17 


.3771 


12.800 


147 


35.92 


.4489 


15.238 


176 


30.00 


.3760 


12.727 


148 


35.67 


.4459 


15.135 


177 


29.83 


.3729 


12.666 


149 


35.44 


.4426 


15.033 


178 


29.66 


.3708 


12.684 


150 


35.20 


.4400 


14.933 


179 


29.50 


.3687 


12.514 


150.85 


35.00 


.4375 


14.849 


180 


29.33 


.3666 


12.444 


151 


34.97 


.4371 


14.834 


181 


29.17 


.3646 


12.376 


152 


34.74 


.4341 


14.736 


182 


29.01 


.3626 


12.307 


153 


34.44 


.4305 


14.640 


182.07 


29.00 


.3625 


12.306 


154 


34.28 


.4285 


14.545 


183 


28.85 


.3606 


12.240 


155 


34.06 


.4257 


14.452 


184 


28.76 


.3695 


12.171 


156 


33.84 


.4230 


14.359 


185 


28.54 


.3567 


12.108 


157 


33.62 


.4203 


14.268 


186 


28.39 


.3548 


12.043 


158 


33.42 


.4177 


14.177 


187 


28.24 


.3529 


11.980 


159 


33.21 


.4150 


14.088 


188 


28.08 


.3511 


11.916 


160 


33.00 


.4125 


14.000 


188.5 


28.00 


.3500 


11.883 


161 


32.80 


.4088 


13.912 


189 


27.93 


.3429 


11.852 


162 


32.60 


.4070 


13.859 


190 


27.80 


.3466 


11.789 


163 


32.39 


.4048 


13.741 


191 


27.65 


.3456 


11.727 


164 


32.19 


.4024 


13.668 


192 


27.50 


.3437 


11.667 


165 


32.00 


.4000 


13.576 


193 


27.36 


.3420 


11.606 


166 


31.81 


.3976 


13.494 


194 


27.22 


.3402 


11.646 


167 


31.62 


.3952 


13.41^ 


196 


27.07 


.3384 


11.487 



(242) 



TABLE OF GRADIENTS. 



• 


Ver, dt. 


Ver. alt 


Gnu in 


V^ . m 


Ver. alt. 


Vr.alt. 


Gra. in 


Katio. 


in feet 


in feet 


lbs. doe 


Ratio. 


in feet 


in feet 


lbs. due 


One in 


per mile. 


per cb. 


to 1 ton. 


One in 


per mile. 


perdi. 

.2598 


to 1 ton 


196 


26.938 


.3367 


11.429 


254 


20.787 


8.819 


198 


26.666 


.3333 


11.313 


256 


20.625 


.2578 


8.760 


199.25 


26.50 


.3312 


11.242 


260 


20.30 


.2537 


8.615 


200 


26.40 


.3300 


11.200 


264 


20.00 


.2500 


8.485 


203 


26.00 


.3250 


11.034 


266 


19.85 


.2481 


8.421 


204 


25.882 


.3235 


10.980 


268 


19.701 


.2462 


8.358 


205 


25.756 


.3219 


10.927 


270 


19.55 


.2444 


8.296 


206 


25.631 


.3204 


10.873 


270.77 


19.50 


.2437 


8.273 


207 


25.507 


.3188 


10.821 


272 


19.411 


.2426 


8.235 


208 


26.38 


.3173 


10.769 


274 


19.270 


.2408 


8.175 


210 


25.14 


.3142 


10.666 


275 


19.20 


.2400 


8.146 


211.2 


25.00 


.3125 


10.606 


276 


19.17 


.2396 


8.116 


212 


24.905 


.3113 


10.666 


278 


19.00 


.2375 


8.057 


216 


24.44 


.3055 


10.370 


280 


18.85 


.2366 


8.000 


220 


24.00 


.3000 


10.181 


282 


18.72 


.2340 


7.943 


224 


23.571 


.2946 


10.000 


285.4 


18.60 


.2312 


7.849 


225 


23.47 


.2934 


9.955 


286 


18.461 


.2307 


7.832 


226 


23.362 


.2920 


9.911 


288 


18.33 


.2291 


7.777 


228 


23.158 


.2895 


9.824 


290 


18.20 


.2276 


7.724 


229.5 


23.00 


.2875 


9.760 


292 


18.08 


.2260 


7.671 


230 


22.95 


.2868 


9.739 


293.3 


18.00 


.2250 


7.637 


232 


22.758 


.2844 


9.655 


296 


17.838 


.2229 


7.567 


236 


22.373 


.2796 


9.491 


300 


17.60 


.2200 


7.466 


240 


22.00 


.2750 


9.333 


301.71 


17.50 


.2187 


7.424 


242 


21.82 


.2727 


9.256 


306 


17.22 


.2152 


7.320 


245 


21.55 


.2693 


9.143 


310 


17.03 


.2128 


7.226 


246 


21.46 


.2682 


9.105 


310.6 


17.00 


.2125 


7.212 


250 


21.12 


.2640 


8.960 


314 


16.812 


.2101 


7.134 


251.4 


21.00 


.2625 


8.910 


315 


16.76 


.2095 


7.111 


262 


20.952 


.2619 


8.888 


318 


16.603 


.2076 


7.044 





(S4J) 


ll 




TABLE OF GRADIENTS. 


lUtio. 


TcT.i]t.lT<r..L'Gn.B 




Vff. dt 


V^..I.<G» i. 


infcct jnfa4 iIlM.dw 
^Bik.jMr<L|toItni 

16.50 1.206? 7.000 


BMio. 


ikln 


mkt 


Ow-iIk 


Om in 


OMm 


OTldk. 


^^ 


mlm 1 


320 


t06.1 


13.00 


.1625 


5.517 ' 


322 


16.398.204+! 6.956 


410 


12.878 


.1609 


5.463 


326 


16.196.2024 6.871 


415 


12.722 


.1590 


5.397 


330 


16.00 .2000.6.787 


420 


12-57 


.1571 


5.333 ■ 


334 


15.809.1976,6.706 
15.621i .19521 6.627 


122.4 


12.50 


.1562 


5.303 1 


338 


425 


12.42 


.1553 


5.230 1 


340 


15.53 


.1941 6.58* 


(30 


12.28 


.1535 


5.209 


344 


15.34 


.1917 6.511 


435 


12.13 


.1517 


5.149 


348 


15.122 


.1896 6.436 


440 


12.00 


.1500 


5.090 


350 


15.08 


.1885 6.400 


445 


11.865 


.1483 


5.033 


352 


15.00 


.1875 6.363 


449 


11.76 


.1470 


4.989 


356 


14.831 


.1854 6.292 


450 


11.73 


.1466 


4.977 


360 


14.666 


.1833 


6.222 


455 


11.626 


.1453 


4.923 


362 


14.589 


.1823 


6.188 


459.13 


11.50 


.1437 


4.879 


364.13 


14.60 


.1812 


6.152 


(60 


11.47 


.1434 


4.869 


366 


14.426 


.1803 


6.120 


(65 


11.35 


.1419 


4.817 


370 


14.27 


.1783 


6.054 


(70 


11.23 


.1404 


4.766 


372 


14.192 


.1774 


6.018 


(75 


11.12 


.1380 


4.716 


375 


14.08 


.1760 


5.973 


(80 


11.00 


.1375 


4.666 


376 


14.04 


.1755 


5.957 


(84 


10.909 


.1363 


4.628 


377.1 


14.00 


.1750 


5.941 


(85. 


10.886 


.1361 


4.616 


380 


13.90 


.1737 


5.894 


(90 


10.77 


.134( 


4.571 


384 


13.75 


.1718 


5.833 


(95 


10.666 




4.525 


388 


13.608 


.1701 


5.773 


500 


10.56 


1320 


4.480 


390 


13.54 


.1692 


5.743 


502.85 


10.50 


:i312 


4.455 


•Ifti 11 


13.50 


IA»7 


.1797 


=i05 


10 4'i'^ 







1 



(244) 
TABLE OF GRADIENTS. 





Ver. Alt. 


Yr. ALIGrs. inl 


V% M.9 


Ver. Alt 


Ver.Al. 


6n.in 


Ratio. 


in feet 


infect 


ll».dae 


Ratio. 


in feet 


in feet 


lb8.diie 


One in 


per mile. 


per di. 


tolton 


One in 


per mile. 


perch. 


tolton 


520 


10.15 


.1268 


4.307 


690 


7.65 


.0956 


3.246 


525 


10.057 


.1257 


4.266 


700 


7.54 


.0942 


3.200 


528 


10.00 


.1250 


4.242 


704 


7.50 


.0937 


3.182 


630 


9.962 


.1245 


4.226 


710 


7.436 


.0929 


3.155 


635 ^ 


9.86 


.1233 


4.187 


720 


7.33 


.0916 


3.111 


540 


9.77 


.1222 


4.148 


730 


7.23 


.0905 


3.068 


545 


9.689 


.1211 


4.110 


740 


7.13 


.0891 


3.027 


550 


9.60 


.1200 


4.072 


750 


7.04 


.0880 


2.986 


555 


9.513 


.1189 


4.036 


754.3 


7.00 


.0875 


2.970 


555.68 


9.50 


.1187 


4.031 


760 


6.947 


.0868 


2.947 


560 


9.42 


.1178 


4.000 


763 


6.92 


.0865 


2.935 


565 


9.345 


.1168 


3.964 


770 


6.857 


.0857 


2.909 


570 


9.266 


.1150 


3.929 


780 


6.77 


.0846 


2.871 


575 


9.182 


.1145 


3.895 


790 


6.685 


.0835 


2.835 


580 


9.103 


.1138 


3.862 


800 


6.60 


.0825 


2.800 


586 


9.02 


.1128 


3.829 


810 


6.518 


.0814 


2.765 


686.6 


9.00 


.1125 


3.818 


812.3 


6.50 


.0812 


2.758 


590 


8.949 


.1118 


3.796 


820 


6.439 


.0805 


2.731 


595 


8.874 


.1109 


3.764 


830 


6.36 


.0795 


2.698 


600 


8.800 


.1100 


3.733 


840 


6.285 


.0785 


2.666 


610 


8.655 


.1082 


3.672 


850 


6.21 


.0776 


2.635 


620 


8.516 


.1064 


3.611 


860 


6.139 


.0767 


2.604 


621.17 


8.50 


.1062 


3.606 


870 


6.06 


.0758 


2.574 


625 


8.48 


.1056 


3.584 


880 


6.00 


.0750 


2.545 


630 


8.38 


.1047 


3.555 


890 


5.932 


.0741 


2.516 


640 


8.25 


.1031 


3.500 


900 


5.86 


.0733 


2.488 


650 


8.123 


.1015 


3.446 


910 


5.802 


.0725 


2.461 


660 


8.00 


.1000 


3.393 


920 


5.739 


.0717 


2.434 


670 


7.880 


.0985 


3.343 


930 


5.67 


.0709 


2.408 


680 


7.76 


.0970 


3.294 


940 


5.617 


.0702 


2.383 



« 




(245) 




TABLE OF GRADIENTS. 


V^ J.* 


Ver.Al. 


Ver.Al. 


6m. in 


V^ . • 


Vr.Al 


Vr.Al. 


Gnuin 


Ratio. 


in feet 


in feet 


lbs. doe 


Ratio. 


in fee 


infbet 


lbs. due 


One in 


per mile 


perch. 


to! ton 


One in 


per m 


perch. 


to 1 ton 


950 


5.55 


.0694 


2.357 


1760 


3.00 


.0375 


1.272 


960 


5.50 


.0687 


2.333 


1800 


2.933 


.0366 


1.244 


970 


5.443 


.0680 


2.309 


1850 


2.854 


.0356 


1.211 


980 


5.38 


.0673 


2.285 


1900 


2.780 


.0345 


1.178 


1 990 


5.333 


.0666 


2.262 


1950 


2.707 


.0338 


1.148 


1000 


5.280 


.0660 


2.240 


2000 


2.64 


.0330 


1.120 


1050 


5.028 


.0628 


2.133 


2100 


2.514 


.0314 


1.066 


1056 


5.00 


.0625 


2.121 


2112 


2.50 


.0312 


1.060 


1100 


4.80 


.0600 


2.036 


2200 


2.40 


.0300 


1.018 


1150 


4.591 


.0574 


1.947 


2300 


2.295 


.0286 


0.973 


1173.33 


4.50 


.0562 


1.909 


2400 


2.200 


.0275 


.933 


1200 


4.40 


.0550 


1.866 


2500 


2.112 


.0264 


.896 


1234 


4.2787 


.0535 


1.815 


2600 


2.030 


.0253 


.861 


1250 


4.224 


.0528 


1.792 


2640 


2.00 


.0260 


.848 


1300 


4.061 


.0507 


1.723 


2700 


1.955 


.0244 


.829 


1320 


4.00 


.0500 


1.697 


2800 


1.885 


.0235 


.800 


1350 


3.911 


.0488 


1.659 


2870 


1.84 


.0230 


.780 


1400 


3.77 


.0471 


1.600 


2900 


1.820 


.0227 


.772 


1440 


3.66 


.0458 


1.555 


3000 


1.760 


.0220 


.746 


1450 


3.641 


.0455 


1.544 


3200 


1.650 


.0206 


.700 


1500 


3.52 


.0440 


1.493 


3400 


1.553 


.0194 


.658 


1508.57 


3.50 


.0437 


1.485 


3520 


1.500 


.0187 


.636 1 


1520 


3.47 


.0433 


1.473 


3600 


1.466 


.0183 


.622 


1550 


3.406 


.0425 


1.445 


3800 


1.390 


.0173 


.589 


1600 


3.30 


.0412 


1.400 


4000 


1.320 


.0165 


.560 


1620 


3.26 


.0407 


1.382 


4285 


1.232 


.0154 


.522 : 


1650 


3.20 


.0400 


1.357 


4500 


1.173 


.0146 


.497 


1700 


3.106 


.0388 


1.317 


5000 


1.056 


.0132 


.448 


1730 


3.052 


.0381 


1.294 


5280 


1.000 


.0125 


.424 


1750 


3.017 


.0377 


1.280 









J 



EXPLANATION 



OF TBI 



USE OF THE TABLES OF GRADIENTS. 

The first column gives the ratio of the horizontal length 
to an nnit of vertical height^ as when in column 1 we find 
160^ it signifies that the gradient indicated has one foot 
vertical rise to 160 feet horizontal lengthy commonly writ- 
ten ^^indination 1 in 160/' The second column gives 
the number of feet of vertical rise per mile due to the incli- 
nation indicated in column 1. The third column gives the 
number of feet of vertical rise per chain due to the same in- 
clination. The fourth column gives the number of poimds 
avoirdupois required to be exerted by an engine to over- 
come the gravitation of one ton weighty to be moved up- 
wards along the same inclination. 

To calculate the rise or fall of gradients^ multiply the 
tabular number corresponding to the required chanu^er 
opposite to the given inclination by the lengthy and the 
product will give the total rise in feet. Example — Gra- 
dient 1 in 160^ length 125 chains. Opposite 160 we find 
.4125 j and .4125 x 125 = 51.56 feet vertical rise. 

It is hardly necessary to observe that these tables ad- 
mit of being applicable to gradients not contained therein, 
by applying any required multiple — as 2, 8, 4, 5, &c., or 
h if if h ^' — ^ some gradients included in the table. 
For instance^ the rise due to 1 in 19 may be found by 
doubling that of 1 in 38^ and the rise due to 1 in 4650 
may be found by dividing that due to 465 by 10, or that 
of 155 by 30, &c. 



While plotting a series of gradients, I sfaoold sDggest 
the following form of arranging them, as rendering the 
calculations easier to be checked and to be referred to : — 



LIST or GaADIBNTS, 



_ at a point elevated {50) feet above the 
line: — 



Ineliiiatian. 


DMdibc 

whether 
iMorfUl 


r 


otal length 
terminiu. 


Veit-tiM 


Vert.Wl, 


Nl 






nLdlw. I 


■Jlee 


ehni. 


feet. 


feet. 


fest. 


1 in 160 


rise 


40 




40 


16.50 





66.50 


252 


f«n 


32 




72 





8.38 


58.12 


4801 fall 


70 


1 


62 





9,42 


48.70 


Level level 


45 


2 


27 








48.70 


132 ri.e 


62 


3 


9 


31.00 





79.70 


4M rise 


18 


3 


27 


2.56 





83.26 


920 fall 


1 11 


4 


38 





6.52 


75.74 


100 Ml 


28 


4 


66 





1.85 


78.89 


264 rise 


4« 


5 


82 


11.50 




85.39 


Proof of Length | 5 82 






61.56 1 26.17 1 
26.17 


Effective nse 






35.89 


Add terminnB above datum — 




50.00 


Proof ofl 


leight 


- 






85.39 







By computing the gradients from the total lengths, they 
will be made to agree with the mileage. I recommend 
that the gradients be estimated in feet and decimals, and 
not in feet and inches, the decimal method being more 
accurate. 

The use of the fourth column in the table 
described — 

Suppose that it were required to know tli 
influence of any incline on a railway — for exi 
incline of 1 in 160 ; then in column four, < 
we find 141bs, per ton; and if the load 



248 

(including the engine and tender) be 100 tons, then 
100 X 14 = 14001bs. would represent the resistance to 
motion upwards^ and also it would represent the force 
assisting the train to descend the same gradient. It is 
manifest that the forces up and down are in equilibrium ; 
therefore any given incline will not, by itself, be a suffi- 
cient datum to find the preponderating influence of the 
whole series of inclines. By the term preponderance I 
wish to express " the excess of force due to gravity alone, 
with which the load drawn resists the tractive power, in 
one direction, over that force which is similarly exerted 
in the opposite direction.^^ As the resistance, due to 
gravity, on an ascending gradient, is measured by the 
height through which the load is raised ; and as the pro- 
petting force, due to gravity, on a descending gradient, is 
measured by the height through which the load is lowered; 
it follows, that part of the force expended to raise the load 
wiU be restored by the self-propelUng influence of gravity, 
when the load shall begin to descend. If the descent is 
greater than the ascent, there will be a surplus of gravi- 
tating power, or the "preponderance" will be active in 
the opposite direction to the given motion; but if the descent 
is less than the ascent, there will remain a portion of the 
tractive force unbalanced, or the ''preponderance" vrill be 
active in the given direction ; and, in either case, its retard- 
ing action is always due to the difference of the ascent and 
descent. Therefore, in a series of inclines, some rising 
and some falling, the effect in any given direction will 
be proportional to the difference of the sum of all the 
ascents and the sum of all the descents in that direction ; 
therefore if it were required to know the preponderance of 
the series of gradients given in the preceding form, it 
win be necessary to multiply the corresponding tabular 
number found in the fourth column by the length, and 
setting the ascents in one result, and the descents in 
another, and dividing their difference by the total length, 
the result will give the preponderance in pounds per ton 



in favour of that series of gradients irhose sam is greatest. 
The following form is adapted for the purpose : — 





Describe 




Grarity ot 


Product 


Product 


Indiiutlon. 


whether 


Lo«h. 




of 


of 




lieeorfBU. 


chaiu. 


in lbs. 


rijoo. 


fBlis. 


1 in 160 


lise 


« 


14.00 


560. 





252 


faU 


32 


8.89 





284.48 


490 


faU 


70 


4.67 





319.90 


Level 


level 


45 











132 


rise 


62 


16.97 


1052.14 





-l&l 


rise 


18 


483 


86.94 





920 


m 


91 


2.43 





221.13 


100 


M 


28 


22.40 





62.72 


261 


rise 


46 


8.48 


390.08 


— - 



2089.16 888.33 



Divide by the Length 



4p32) 1200.93 



Preponderance due to the rises in poimd8=2.78. 

The same result may he obtained in a less complex 
manner, hut the shove form ^ves the ratios each separate 
length of gradient hear, when compared with the whole 
length; for instance, -7^ = 1.396 is the preponder- 
ance in ponnds per ton due to the rise of 1 in 160; that is, if 
1 in 160 were the only gradient, uid the rest of the hue 
were a level, then 1.396Ibs. per ton repr 
of power required over the whole line, 
length of the gradient only ^ 40 chi 
gravitation is 14db3. per ton; and the 1 
individual preponderances due to the risei 
those due to the falls, will give a total of 
required in the given direction to overci 
innue'tace of gravitation. When the trai 



250 

the same series of gradients, the preponderance is in 
favour of the motion — ^that is, assists it ; because the rises 
become falls and the falls rises. Hence the excess of 
power required in the one direction, compared with that re- 
quired in t/ie other, will be 2.78 x 2 = 5.561bs. per ton. 
In order to compute the total excess at once without refer- 
ence to the intermediate gradients, it is sufficient to know 
the eflEective rise between one terminus and the other, and 
the total length. Let H be the effective rise in feet, and 
let L be the total length in chains ; then the rule for the 

excess is -— -j; — ; and taking, for example, the preceding 

data, we have H = 35.39 and L = 432. The excess in 

this case, therefore, is —^ — -^ — '- — = 5.551bs. per ton, 

the same as above. If the length is given in milesy the 

rule for the excess becomes —^ — , and the above example 

will work out - — -^ — '— = 5.551bs. per ton. 



For the convenience of those who prefer a literal, instead 
of an algebraical rule, I subjoin the following : — 

Rule, to find the excess of power required per ton in one 
direetion, compared with that required in the other — 

Ist. When the " effective rise'^ in feet, and the " total 
length'^ in chains are given — 

Multiply the rise by 67.86, and divide the product by 
the length: the quotient is the required ^^excess^^ in pounds 
per ton. 

2nd. When the ^^ effective rise^' in feet, and the " total 
lengtV^ in miles are given — 

Multiply the rule by 0.848, and divide the product by 
the length : the quotient is the required ^' excess^^ in 
pounds per ton. 



ESSAY 

ON THE POWER REQUIRED TO OVERCOME 

THE RESISTANCE ON INCLINED PLANES. 



As the Table of Gradients above described is not saffi> 
cient to enable an estimate to be formed of the total 
power required to put in motion a given weight of train 
with a given- velocity, it is to approximate this object that 
the following few pages are devoted. 

The ordinary resistance to which a train is subjected 
during motion may be separated into three kinds : 

Istly. The resistance due to the frictiou of the carriages 
and of their load. 

2ndly. The resistance due to the air acting on the 
surfaces of the carriages. 

Srdly. The resistance due to the gradient. 
I.— Of the Friction. 

Eiperience has determined that, for all carriages con- 
structed on the same principle and with the same dimen- 
sions of hearings, gauge of rails, and general construction 
of parts, the firiction is a conatant quantity for all veloci- 
ties, and is expressed by a certain proportion of the weight 
drawn. Therefore, if the symbol / denote this ennHtniit 
proportion, and W the weight drawn, 
pressing the total friction of the train. 
If one carriage with its load weighs 6 tc 
10 carriages, and if the proportion whic 
tained by experiment is taken at 711 
(/=7) X (W=6k10) = 420 lbs., which ii 
- ance due to friction. But> in estimati 



252 

be overcomej besides the carriages^ the engine in fronts 
being of a different construction^ should be taken into 
consideration. Some experiments have shewn that the 
friction of an 11-inch cylinder engine whose weight is 
about 8 tons^ when running on 4 wheels^ is about 13 lbs. 
per ton ; and a 14-inch cyUnder engine on 6 wheels^ the 
weight being about 11 tons^ offers a resistance by friction 
of 16 lbs. per ton. Supposing that an engine of the first 
description were used, it would be proper to add to the 
preceding calculation 8x13=104 lbs., making a total of 
420 + 104 = 524 lbs. ; and if a 6-wheel engine were used, 
as above described, 11 xl6=176 lbs. must be added, making 
a total of 420+176=596 lbs. : the engine tender is here 
supposed to be one of the carriages. But the above 
allowance for engines must necessarily vary with their 
construction and weight. 

In general, for approximation, 4 carriages maybe added 
for a 4-wheeled engine and tender, and 6 carriages for a 
6-wheeled en^e and tender, and the gross weight of each 
carriage (of the passenger class) may be taken at 5 tons ; 
but this approximation must be understood to refer only to 
a railway laid down with a 4 feet 8 J inches, or 5 feet gauge. 

Hence, if n expresses the number of carriages actually 
containing passengers or goods, the total resisting load for 
a 4-wheeled engine is 5 (n + 4) tons, and the friction will 
te (/= 7) X (W = 5 » + 20) in pounds. 

II. — Of the Resistance of the Air. 

The resistance due to the medium through which the 
train pusses is divided into two parts — one being that due 
to the direct or perpendicular surfaee exposed to th^ 
mediupi in the direction of the motioa^ ; and the other is 
due to the lateral action of the particle? of air on the sides, 
top, and liottom of the carriages. 

First — To treat of the direct surface. 

This is measured by the actual transverse area of the 
railway carriages, with their wheels, springs, and frame- 



253 

work, such as would be represented on a geometric plane, 
wliich is commonly called hj draughtsmen a " transverse 
elevation" of the object. This area, then, is part of the 
direct surface exp(»ed. Another portion is computed bv 
supposing that the extreme mobility of the air enables it 
to intersect and oppose itself to all the various openings, 
comers, and parte not represented by the "trausvcrse 
elevation," but which by their situation would be so if a 
section were mad^ so as to expose them in front ; instead 
of which, the air has access to them by tortuous and 
indirect channels. With these parts must be included 
the front surface of every succeeding carriage to the first, 
as there is a large space between two carriages which very 
freely permits the air to act on each carriage in succession. 
From the immeuse variety of the dimensions and position 
of the parts, it is quite impossible to arrive at any thing 
more than an approximate measure of the surface ; but if 
this be chosen with discretion, the error will not materially 
affect the accuracy of the estimate. 

Having ascertained the sur&ce, the nature of the resist- 
ance is next to be considered. The motion of trains 
through the air is so very variously influenced by the state 
of the medium, in regard to its barometric and tbermo- 
metric condition, and also in regard to the direction and 
force of the wind at the time, which again is varied by 
the winding course of the railway, that it is also on this 
account impracticable to give a true solution to this prob- 
lem. Therefore, in order to fix a standard to which these 
calculations shall refer, let it he assumed that the air has 
a constant barometric pressure of 30 inches, or 14.706 lbs. 
per square inch nearly. The temperature is to be 60<» 
Fahrenheit, and the air to be quite calm or at rest. Then 
the resistance of the air will be measured by 
force which the air itself would exert agains 
surface if moving against it with the given velocit 
over, since both theory and practice have establ 
the resistance of air in motion varies as the 



254 



the velocity directly^ we derive the resistance offered to the 
carriages when we know the resistance dne to some given 
velocity. Thus^ if F is a given velocity^ and A is the 
resistance due to it for a square foot of surface^ and V is 

AV* 
any other velocity, then, asFa:A::V: ^=aV* = 

the resistance for any velocity when a = p • 

With regard to the total surface of the train in direct 
opposition to the air; let S represent the transverse 
area, and include in this some allowance for the surfaces 
of various parts indirectly exposed to the air. Also let Z 
represent the firont surface of each succeeding carriage 
beisides the first. This expression Z will be less than the 
transverse area (S) by a quantity due to the obliquity of 
direction of the impinging particles on the plane of the 
carriage. This obliquity will depend on the variation of 
distance between the carnages, and of their breadth 
according to their construction. The following investi- 
gation is suggested as a means of arriving at some definite 
proportion of the whole to the effective surface. 

In Fig, 1, let ABCD be the surface of the back. 



Rg. 1. 




and let there be another in 
front of it of equal transverse; 
and also let AE be the distance 
between them. 

By first considering the ac» 
tion upon a simple plane only 
in the direction of the motion. 



In Fig. 3, the line AB shews the extreme limit of the 



action of the uninterrupted 
column of air rushing in from 
the opening AC ; and the line 
A C shews the other lunit for 
the same opening : also each of 
these lines may respectively 
t^present the full force of the 




255 

air. Any intennediate line (AD) representa the force of 
the air from one aide at the point J). In like manner 
EC ED EB represent the force of the air upon the points 
C, D, B, trom the other side EB ; tiierefore, the full action 
on any point (D) is the joint action of the forces AD ED, 
and by completing the parallelogram AFED the line 
FD represents the force acting on the point D. But 
because AFED is a parallelogram, AE ia bisected in G 
by the diagonal FD, therefore the force on D is twice GD, 
and also GC, GB, represent half the forces on C and B, and 
generally the whole effect of the air on the surface CB 
will be twice the sum of all the forces GD. Hence, in 
Mg. 1, iff is a centre point in the body of a carriage 
adjoining^e surface ABCD, the bundle of forces directed 
against ABCD will be contained in the pyramid FABGD, 
whose vertical axis is S AE. 

InFigS, letiAD=AC = A, letEC = «,widBC=£, 
FiR. 3. the force EB is /= Va* + «>. 

- . A J But by reducing any force (EB) 

f g ^ to its respective horizontal and 

I e l— -'-'^ g 1 vertical forces, the intensity of 

this force is diminished in the 
ratio of the 4th power of the 

cosine of the impinging an gle; the refore, at any point (B), 
the radius being EB = Va= + «", and the cosine EC = a, 






u; 



the force -/a" + a^ becomes IT^^^' 

The mean radius is EB = "-^^■^^+? nearly, therefore 

the effective mean force becomes for the whole surface 

,-,f^|^»d by putting J-., 

the whole effect will be for each foot of s 



266 



The rule I have adopted in computing the effective surfiEU^ 

^^ fmr~~7=^n when n is equal to the full breadth divided 
(1 + 2 vi+n*) • ^ 

by twice the intermediate space ; and computiBg the pro- 
portions due to the above rule^ we have — 



When n = 

€t 
tt 
€C 
€i 



1 the effective surface is ^ 



7 

8 

8 

4 
5 

8 

1 
2 



€€ 



€t 



a 



t( 



6 

ao-* 



of the whole. 
it 

u 
cc 
tt 



Assuming the third proportion to be nearest to the 
practice in construction, and adopting this figure in the 
estimate, the expression Z becomes S x ^, and for all the 

carriages less the first *^*"" ^ expresses their resisting 

surfiace; and adding the engine and tender, whichalso receive 
a share of the direct action of the air, the total resistance is 

found in the formula — ^ — - x a V*. 

It now remains to give some expression for the lateral 
resistance of the air. Let K represent the actual measure- 
ment of a railway carriage, such as would be represented on 
a drawing denominated a '^longitudinal elevation^^ of the 
object, and to this surface add some allowance for those parts 
which are likewise indirectly exposed to lateral action. 
For all trains in motion there must necessarily be a current 
of air directed against the sides of the carriages, caused 
by the eddying motion given to the air displaced in front 
by the forward motion, which then acts along the sides of 
the carriages in its endeavours to reinstate itself. This 
action of the air cannot be accurately appreciated ; yet 
some accoimt should be taken of it, as, however trifling it 
may appear, the surface exposed to it is very great, varying. 



257 

in truss of 10 or 15 carriages, from 5,000 to 10,000 aqnare 
feet. Then let / represent tbe resistance due to one 
square foot. / E (n + 2) is the whole lateral resistance ; 
and the whtAB resistance lateral and direct is ex- 
pressed hy riiilJL« + K/(n+2)] x V, and making 

Q = ' '" 3 ^^ " + K / (» + 2) it hecomes QV. 

It will he proper now to give some value to the coefS- 
cients a and /. The first is foiind hy the well known laws 
of pneumatics. The height of the nniform atmosphere 
which surrounds the earth is computed to be, on an avenige, 
26,400 feet. Eight times the square root of this hei^t 
will give the velocity with which the air will rugh into a 
vacuum, therefore 8 -/26400 = 1300 feet per second. 
Tliis velocity is due to a pressure of 14.71 lbs. per square 
inch, or 2118 Ihs. per square foot of surface. Hence the 
pressure on a square foot with any other velocity m^ be 
found, because the pressures vary as the squares of the 
velocities. ^ " — = velocity of airin vacuo in miles per 

hour. Now as ^^°^,' ^^' : 3118 :: V: ^=re8i8tance 
22" 370 

due to any velocity V. Hence a = — - 

By some observations I made in a train moving at the 
rate of 80 miles per hour, the air being quite calm at the 
time, I found that the cnrrent against the carriages was 
acting at an angle of 9 or 9 J degrees nearly, and computing 
the force (which varies as the square of the sine of the 
uigle) we have this proportion — 
> V 1 (sine "Q ) V 

Moreover, this quantity / will vary with tl 
the disturbing force producing the current 
velocity, and therefore the angle varies. 1 



258 

the cause which produces this angle varies as the square 
of the velocity, we have this proportion to find the 
true value of /. Since the angle given is known to 
be due to thirty miles per hour vdodty, then as 

80« = 900: -i~ :: V : -— ^ — = Z. The lateral 

14400 12,960,000 

current of air does not act on the entire lateral surface of 
the train, because the air, after acting on the front surfaces 
of the carriages, is repelled therefrom, and, escaping 
laterally, passes along the carriages in continuous eddies, 
thereby neutralising part of the effect of the currents 
directed against them. No accurate measure can be given 
of the effect produced ; it depends partly on the proportion 
between the lengths of the carriages and the intermediate 
spaces. If we assume that the current begins to act upon 
the carriages after the train has traversed a space equal to 
twice the intermediate distance, and if the length of the 
carriages are taken at a medium to be three times this 
distance, then the side surface effectively exposed to the 
lateral current will not be more than one-third of the 
whole. I therefore, in default of better data, assume this 
proportion; and, by substituting thesevaluesof a and /, in the 

equation of Q,we have Q=tS!tl^ x -L + E!tl2l x — -^— 

^ ' 2 370 3 12,960,000. 

For convenience of calculation, the values of S, K, », 
should be determined, so that every thing relating to an 
estimate of power may refer to a standard. According to 
the existing practice in the construction of carriages, I 
estimate the value of S to be 80 square feet, including the 
allowances for indirect action. The value of K also I take 
at 600 square feet, with similar allowances. With regard 
to n, I assume as a standard 100 tons weight drawn, 
including engine and tender ; and supposing these last 
together weigh 20 tons, the remaining 80 tons divided by 
five (the average tonnage per carriage) will give 16 car- 
riages, or 131 = 16^ 



259 

Substitating these values in tlie equation of Q, we have 
Q _ 8006 + 3) ^6ooMi6+2)V!^ 2 054 ^ JV1_ Hencethe 

2 x 370 38,880,000 3600 

second item of the resistance to motion is 

To fadlitate the application of the preceding rules to 
practical cases^ I subjoin a table which will embrace all the 
different values of S, K, and V, and the variable propor- 
tion (indicated by the symbol w) of distance between the 
back carriages. The general rule for the resistance of 

theairis QV3= S^l [(^ + 1) a?+ 1] + KV^ (» + 2) 

370 ^ 38,880,000 

in pounds avoirdupois. 




261 

To estimate the resistance of the air by the tables^ when 
the nnmber of carriages (n) is given^ exclnsiye of engine 
and tender, let the formula become QV* = SN + K (» + 2), 
Look in table S for the number corresponding to the 
given velocity and &ont surface, and in table N for the 
number corresponding to the number of carriages (n), and 
to the given proportion of the intermediate space to the 
breadth of the carriage {a:), and multiply the tabular num- 
bers found together; and to this product add the product 
ofn + 2, and of the tabular number in Table K correspond- 
ing to the given velocity, and the side surface of each car- 
riage. This sum will give the resistance required. To 
illustrate this by an example — 

Let t? = 24 miles per hour. 

f< s = 70 feet front surface. 

" » = 12 carriages* 

" * = 600 feet side surface. 

'^ a? = half the breadth = intermediate space. 
In Table S, 
Opposite 24, and under 70, we find - - 108.97 

In Table N, 
Opposite J, and under 12, we find - - 5.5 

The product of these^numbers is - - 599.33 

In Table K, 
Opposite 24 and under 600, we find - - 5.12 

and 71+2 = 12 + 2= - - - - 14 



The product ofthesenmnbers is - 71.68 

And ttie sum of 599,33 + 71.68 = - - 671.01 

is the resistance sought in pounds. 

III. — Of the Gradient* 

The force exerted by a body osk BXk inclined plane is to 
the weight of the body as the height c^ the plane is to its 
length ; and this force tends to cause the body to descend 
the plane. Sut if the motion of the train is up the incline, 



262 

then the force acts in resistance to this motion; andif the 
motion is down the incline^ then it acts as an assistant to 
the propelling force applied to overcome friction and 
atmospheric resistance. Therefore^ this term of the equa- 
tion of total resistance is expressed + -- or — ii. when 

,9 9 

g is the ratio of the length of the inclme to its height, 

the term + being used on ascending, and the term — on 
descending inclines. 

Summary of the Resistance. 

From the foregoing investigation, it appears that 
the total resistance to be overcome, to move 100 tons 
along with some velocity (V), along any gradient 
{g) is expressed by the formula, in pounds = 700 + 

2.054 V« + ^ ± 2il!229. And the general rule is 
(5 n . 20)/. *i2-^ r^^^Ll^^ pounds. 

^ ^"^ 740 38,880,000 g ^ 

Also, the equivalent for n in the terms of W, may 
be inserted. Let g express the number found in 
the 4th column of the table of gradients, instead of 

^^% and inserting the previous values of (s) and (K), 
i/becomes Wf 1 Wff . (^T^' . (^-12)1!, or 

46.25 324,000 ' 

W(/- ff -^ 46.25 "*" 324,000/ gjs — 32^400^ ^^^difthe 

velocity were assumed to be at an average of 30 miles 
per hour, and the friction at 71bs. per ton,* the rule 
becomes W (29 ±g)— 122. If the weight were 100 tons, 
the resistance in pounds becomes 2900 + lOOg. Therefore, 
the number 2900 expresses the resistance in pounds of 100 
tons moving along a level plane at the velocity of 30 miles 
per hour, which gives 29 lbs. per ton, or is equivalent to the 
gravity of an incline of one in 77 J. Any incline, therefore, 
whose descent is not in a less ratio than this, will not cause 
the train to accelerate its velocity (being already 30 miles 
per hour) by gravitation. 



263 

SelatioH of the Power required to the Retistanee. 
I will conclude this essay by shewing briefly the relstioo 
existing between the resistance of the load and the power 
of the engines required to draw it along. I resume the 
data upon which the resistance was formerly computed, 
and which are recapitulated as follows : — 
Friction of each carriage, Tibs, per ton. 
Carriages weigh each, 5 tons. 
Engine and tender computed as 4 carrit^es. 
Weight drawn (including engine and tender weighing 

20 tons) 100 tons. 
Front snrface of each carriage (with allowance for 

concealed surface) 80 square feet. 
Side surface of each carriage (with allowance for 

concealed surfaces), 600 square feet. 
Space between carriages, three-quarters the breadth 

of a carriage. 
Velocity of motion, 80 miles per hour. 
The rule for resistance due to the above data is 
3900 ±100 if in lbs. 

To draw a train along a railway, the power of traction 
depends upon the adhesion of the drivmg wheels to the 
rails. This adhesion, therefore, independently of the in- 
clines, must equal SQOOlbs. 

But in estimating the power, it is necessary to take the 
most extreme cases which may occur on the railway, that 
the engine may surmount with its load the steepest inclines. 
In the more recent practice of engineers, the inchnation 
1 in 200 is not un&equently adopted as the standard 
gradient, and to this I will confine myself, as, when inchnes 
are much steeper, the speed is practically considerably 
reduced below 30 miles per hour. On referrin ' " 
table of gradients, opposite 1 in 200, column 4th 
11.20 lbs. per ton, which gives 11.3 x 100 = ] 
for the resistance due to the gradient. 1 
adding together 3900 + 1120 = 4020 lbs. 
required to enable the engine to draw the loa 



364 

given speed. The adhesion varies very much, owing 
partly to the liability of the wheels to slip or skid on 
greasy rails, partly to casual imperfections in the rails 
or machinery, and partlv to inappreciable obstructions, 
oscillations, and irregulanties of motion. It will also vary 
owing to the different manner in which the weight of the 
engine is supported on the wheels, either by the driving 
wheels being coupled or imcoupled, or by there being six 
wheels to carry the weight; also, the diameters of the 
wheels being Afferent cause great variation in the adhe- 
sion; and ako, by varying the construction of the engine, 
the weight imposed on the driving wheels may vary fix)m 

|- to I ofthewhole. 'A combination of the above causes 

produces great uncertainty, to avoid which Ihave fixed as a 
standard, that the adhesion, under the least favourable cir- 
cumstances, of an engine weighing 12 tons, whose weight on 
the driving wheels is 8 tons (being two-thirds of the whole) 
is one-sixteenth part of the weight of the engine when the 
driving wheel is 5 feet 6 inches in diameter, and this pro* 
portion will vary inversely as the diameter of the wheel. 
Supposing that such engines are used, then the adhesion 

of one engine is P ^ J^^^Q = 1680 lbs.; and in order to 

enable a train opposing a resistance of 4020 lbs. to be 
moved at 80 miles per hour along an incline of 1 in 200, 
three engines would be required at least. 

Pursuing the investigation to ascertain the commaxial 
expenditure resulting from the use of the engines to draw 
the load as before given, I have considered that the best 
measure of the effect would be the cubic feet of water 
vaporized under a given pressure in an hour, in preference 
to computing the horses-power, or the lbs. raised 1 foot 
high. By knowing the performance of the boiler as to 
the supply of steam, we can find the real effect produced 
by an engine, when the resistance due to the given unilbrm 
maanmum velocity is determined. 



265 

The quantity of fuel which will cause the vaporization 
of a cubic foot of water, multiplied by the total vaporization 
will give the required commercial eflEect. I will suppose the 
case of an engine with a 12-inch diameter of cylinder 
(whose area is a inches) ; stroke of the piston 1 foot 6 inches 
(the length being / in feet) ; the driving wheel being 
5 feet 6 inches (indicated by the symbol D in feet) ; the 
speed as before being 30 miles per hour (or V miles). 
Then the general expression for the quantity of steam used 
by the two cylinders in one hour, allowing for the waste 

, ., o 4.4/0. 5280 x7V 51.33 /«V 

in the passages, &c., t±i5 x WB^ ^ "HD — 

The relative volume of steam at 60 lbs. per square inch, 
compared "vrith the water generating it, is 880 : 1 ; there- 

fore yg X ^ = .135 ?^ = cubic feet of water vapo- 

rized per hour. Then substituting the values of laY and 
D, we have 124.8 cubic feet of water required per hour for 
one engine, supposing it was fully employed. The actual 
power transferred from the cylinder to the circumference 
of the wheel at the point of contact with the rail is measured 
by the pressure on the piston, reduced in the ratio of the 
length of the stroke to the diameter of the wheel. 

Let (/?) represent the pressure (60 lbs. on the inch), and 
let the previous notation be used, then the general expres- 
sion of the power is yr- for one cylinder only. The ratio 

between the greatest effect of one crank, and the mean 
effect of two acting simultaneously at right angles to each 
other, is nearly as 10 to 16 ; and adopting this proportion, 

we have 1.6 t^ ^ ^ power of two cylinders. Hence, 

substituting the values oi pal and D, we have 2960 lbs. 
nearly for the power of one engine ; and the power of the 
three engines necessary for adhesion will be 2960 x 3 = 
8880 lbs., being 4860 lbs. in excess of the resistance. This 

F F 



266 

excess will be partly taken away by the additional friction 
of two extra engines and tenders^ and the corresponding 
indirect front and lateral atmospheric resistance^ and the 
gravity of 40 tons extra on an mcline of 1 in 200. This 
extra resistance will not exceed 800 lbs., and the whole 
resistance will be 4020 + 800 = 4820 lbs ; therefore, the 
power which can be obtained is more than that which 
is reqnired in the proportion of 8880 : 4820 : : 1.84 : 1. 

To reduce this supply to the demand, we may reduce 
the Tai^ri^tion of 124.8 cubic feet in each boiler by 1.84, 

and then we have ^^'\/ ^ ^ 203^ cubic feet to be actu- 

ally converted into high pressure steam. 

The above computation refers only to the ca«e of an 
ascending gradient; but, of course, it is proper to estimate 
the case of a descending gradient, to obtain a true mean 
value of the effect. Therefore, supposing that the same 
weight is returned along the railway, we have 2900 — 
1120 = 1780 lbs, for the resistance. One engine alone is 
nearly equal to the duty as regards adhesion, which is 1680 
lbs. ; but it is better to use two to be certain of the duty re- 
quired, owing to the variation to which the adhesion is liable. 
The power of one engine being 296jO lbs., two engines would 
exert 5920 lbs., from which suppose 400 lbs. to be deducted 
for the resistance to the extra engine, 5520 is the effective 
power which can be exerted, bearing a proportion to the 
resistance of 3 to 1 nearly. Hence, if the steam is gene- 
rated at the full pressure of 60 lbs., and is allowed to 

expand to 19.3 lbs. pressure in the cylinders, only — 

o. 1 

of the full complement of water is necessary; that is, 
instead of requiring 124.8 x 2 = 249.6 cubic feet, 

^^^ = 80.5 cubic feet are sufficient. Therefore, to go and 

return a journey of 30 miles in one hour, 203.4 + 80.5 = 
283,9 cubic feet of water are required ; and reducing 



267 

this quantity to simple tenus^ we have an average of 
— '— = 9.46 cubic feet of water required per mile for 100 

tons load, when travelling 30 miles per hour, on gradients 
whose constant ascent and descent are 1 in 200. This 
estimate is not exactly applicable to a level line, though 
the aggregate resistance in both cases is nearly the same, 
the ^fference in favour of the level being due to the 
resistance of the extra engine. For on the inclined railway 
we have the resistance 4020 + 800 + 1780 + 400 = 7000; 
on the level we have the resistance 2900 + 400 + 2900 + 400 
= 6600, being 400 in favour of the level, equal to the 
resistance of one extra engine. On a level, the resistance 
being 2900, two engines are necessary ; and the effective 
power being 5520, the proportion of power to the resist- 
ance is 1.9 to 1 nearly. Hence, the steam can be allowed 
to expand from 60 lbs. to 31.6 lbs. in the cylinda*, and 

quantity of water required to be vaporized is ^^^'^ ^ . ^ = 

262.8 cubic feet, that is 8.76 cubic feet per mile, shewing 
.7 of a cubic foot in favour of the level. ' 

It now only remains to shew what duty one engine may 
be put to ; the gradient, velocity, and other proportions 
being the same as before. 

As the adhesion is the limit of the power, and conse- 
quently is the measure of the resistance the engine is 
capable of, it follows that 1680 lbs. is all that is required 
to be exerted by the cylinders. Hence, the resistance 
will bear a proportion to the whole power the engine can 
exert (= 2960 lbs.) of 1 : 1.76 nearly; therefore, expanding 
the steam from 60 to 84 lbs. in the cylinders, the vaporiza- 
tion of ?^^ = 70.9 cubic feet of water is the limit of effect 
1.76 

required. Also we have this equation of the power 
required given in terms of the number of carriages drawn 

(/ = 7) X (5« + 20) H- ('■—= 194.59) x Q~±^) 

F F 2 



268 



( 



KV« 



38,880,000 



= 12.5) X (« + 2) + (^ = 11.2) y (5» + 20) 



- 1680, and reducing this equation n = '-^^^' • 

This will give 16 carriages^ equal to 16 x 5 + 20 = 100 
tons descending, and 5 carriages, equal to 5 x 5 + 20 = 45 
tons ascending ; also, we deduce that the engine can draw 
eight carriages =8x5 + 20 = 60 tons on a level. 

For the convenience of those who wish to calculate the 
vaporization required with pressures in the boiler different 
from the assumed standard of 60 lbs., I subjoin the 
following table of relative volumes of the steam compared 
with the water generating it : — 



Presmre per 


BelatiTe Volnine 


iquareinch 


of Steam, 


above 


that of water 


atmoflphere. 


being 1. 


70 


340 


65 


359 


60 


381 


55 


406 


50 


434 


45 


467 


40 


506 


35 


552 



Water vaporized 

per hoar at maximum, 

power of Engine. 



cubic feet. 

139.6 
132.2 
124.5 
116.9 
109.3 
101.6 
93.8 
86.0 



Maximum power 

of one engine, at the 

point of contact 

with the rail. 



lbs. 

3453 
3206 
2960 
2713 
2466 
2220 
1973 
1727 



Prom the above table it is easy to determine any modi- 
fication of the preceding calculations the reader may think 
fit to make. 

It is necessary to state here, in concluding this portion 
of the essay, that so much does the capability of an engine 
to draw a load depend upon the state of the rails, that the 
preceding estimate of adhesion = 1680 lbs. may be con- 
sidered as very low ; indeed, experiments and observations 
have been made, showiiig that the adhesion varies £rom 

-th to ^th of the weight on the driving wheels ; this 



269 

great variation being consequent upon the temporary 
circumstances of weather, cleanliness, and state of repair 
of the metals ; so that, since the preceding calculatioi^s are 
founded upon the lowest datum, the capability of an engine 
may possibly be increased two or three times that which is 
indicated by the results I have given. It appears that the 
cylinders are capable of exerting a power equal to 2960 lbs. 
with steam of 60 lbs. pressure ; therefore, in ordinary cir- 
cumstances, the adhesion may be estimated as equal to this 
power. This will give an increase of effect produced much 
above the calculated effect ; for, by estimating the term 
1223.12, in the equation of n, to be increased by adding 
to it (2960—1680 =) 1280, and therefore making this 
term = 2503.12, we shall find that 

A descending train may be = 150 tons. 

An ascending train may be = 85 tons. 

A train on a level may be = 105 tons. 
The boiler, in this case, must vaporize its maximum 
quantity of steam. It is also obvious that, under very 
favourable circumstances, the adhesion maybe much greater 
than even this estimate; so that by increasing the pressure 
of the steam a greater velocity can be obtained, or by 
enlarging the diameter and stroke of the cylinders much 
greater loads may be drawn. Further, if instead of em- 
ploying one pair of driving wheels, the two pair of a four- 
wheeled engine are coupled, then the whole of the etigine 
is exerting its weight to produce adhesion. In the case 
which I have assumed, the engine weighs 12 tons, with 
8 tons resting on one pair of wheels ; but if the machinery 
is so constructed that 6 tons may rest on each pair of 
wheels, then the adhesion will be increased one-half more ; 
or the tractive force, instead of 2960 lbs., may be estimated 
at 2960 + 1480 = 4440 lbs. Therefore, the term 2503.12, 
in the equation of n as last determined, may be increased 
by 1480, which makes it = 3983.12. Hence we shall 
find, that, in ordinary cases, a four-wheeled coupled engine 
is capable of drawing 



270 

A descending train = 230 tons. 

Or an ascending train = 125 tons. 

Or a train along a level = 155 tons. 
The reader is requested to bear in mind, that the num- 
bers here given are only comparative, and that they depend 
entirely upon the data assumed in the foregoing pages. 
In consequence of the many varying circumstances of con- 
struction and other conditions, I am compelled to give a 
general illustration only. It is not withm the scope of 
this essay to enter into further particulars, which more 
properly belong to a treatise on locomotive engines ; and 
therefore, such readers as desire to obtain a knowledge of 
the details which govern the mechanical action of an engine, 
will find the subject amply discussed in works ably written 
by the Comte de Pambour, Mr. Wood, and others. 



ACCIDENTAL CAUSES OF RESISTANCE. 

The resistance of a train as brought forward in detail in 
the course of this essav is all that can be taken into account 
when forming a general estimate ; but there remain two 
important accidental causes of resistance, which it is not 
improper now to mention. The first cause which I shall 
draw attention to is the resistance opposed by the wind. 
The second cause is the resistance opposed by the journey 
of a train along curves on the Kne. 

The Resistdnce of the Wind. 
I have hitherto considered that the air through which 
the train passes is quite calm, or at rest. But this is very 
seldom the case; and though, when gentle winds are 
blowing, the effect on the surface of the train is not very 
remarkable, yet strong winds will produce a sensible retarda- 
tion of velocity; or an additiontd tractive force is neces- 
sary to overcome the resistance of the wind, and maintain 
the required velocity nnimpaired. As the wind may arise 
from any point of the compass, but the direction of the 
railway is fixed, the wind will impinge, generally, in an 



271 

obKque direction upon the surface of the train. It is, 
therefore, necessary to determine such a general rule as 
shall meet any given circumstances of direction. 

Suppose that the wind acts upon the train at some given 
angle contained between the directions of the wind and 
train respectively. The force which the wind will exert in 

its true direction is ^, when Y is the wind's velocity in 

miles per hour. This diagonal force can be resolved into 
two forces; one acting perpendicularly and the other 
parallel to the direction of the train. That which is per- 
pendicular is proportional to the square of the sine of the 
given angle, and the force is represented by the expression 

sine* X —-. This force, whichever waythe train is moving, 

is constantly soliciting it to act against the side of the 
opposite rail, and tends to produce friction. The force 
which acts parallel with the train^s direction is proportional 
to the square of the cosine of the given angle, and is repre- 
sented by the expression cos.2 x Jl . This force, accord- 
ing as its resolved direction and velocity is opposed to, or 
composed with, the direction and velocity (rf the train^ 
requires an increase of the tractive force, or becomes itself 
a propeller. The symbol Y expresses the absolute velocity 
of the wind ; but since trains in motion are now under 
discussion, the expression above given will indicate only 
the effect produced on a train at rest. The real effect 
of the wind is due to its relative velocity. If the wind 
and train are moving in the same direction, the wind can 
produce no effect unless Y = V = velocity of the train* 
Therefore, in this case, V — Y = Z is the effective velocity 
which aids in propelling the train when V is less than Y ; 
and V + Y = Z is the effective velocity which opposes 
the train when the wind and train move in opposite direc- 
tions. Also, when V is greater than Y, and the wind 
moves with the train, the effective velocity is V — Y = Z^ 
which wiU oppose the train. But in the fbrmer part 



an 

of this essay we have already estimated tlie effect due 
to V^ (corresponding with the velocity of the train) ; 
therefore, the expression which will correspond with the 
velocity of the wind only, is Y' ± 2 VY = 7?. The sign — 
being used when the wind and train move together, and 
the sign + when in contrary directions; in both cases 
acting in opposition to the train^s motion. Also, the 
sign — , is used when the wind and train move together; 
and the term Y being greater than Y, the wind assists in 
propelling the train. Therefore, the general expression, 

cos.» X i-. will now become cos.^ x - = cos.' — . 

371 371 371 

All winds may be considered as blowing in a horizontal 
direction ; therefore in estimating the amount of surface 

..^ .. the W exp^ssed b, .ine- ^^. i. U .ot 

necessary to estimate the tops, bottoms, and the reverse 
sides of the carriages ; and the opposed surface will be 
sufficiently taken, if the expression K, which represents 

the side surface, be made — : then the resistance due to 
the wind acting on the sides of the carriages of thewhole train 
will be indicated by the expression sine^ x M5_2) ^ Y ^ 

The surface exposed to the force expressed by cos.^ x -_ 

varies according to the angle of direction of the wind, and 
to the intermediate distances between the carriages. When 
the angle does not exceed a certain number of degrees, 
the whole front surface of the carriage is not exposed to its 
action, part of it being sheltered by the adjoining carriage. 
On calculating these limits due to the different intermediate 
spaces, I find that when the intermediate distance 



Is equal to the breadth. 


00 


45 deg. min. 


Is 7-eighths do. 




48 « 48 « 


Is 3-fourths do. 




53 " 7 " 


Is 5-eighths do. 


a; 


57 " 59 " 


Is 1-half do. 


^ 


63 " 26 " 



273 

fience, an; angle wliicli is less than tlie angle due to tlie 
given intermediate distance will not permit the whole front 
Bur&ce to be acted upon by the wind. If the front surface 
be espresaed by S as used in a former part of this essay, 
and X express the fraction of the intermediate space when 
the breadth of a carriage is — I, the actual surface exposed 
will be SX X — ', and the resistance mil be expressed by 

^^ — ^^, and for the whole train it becomes 



SZ' (3^ + it^) ^^'-"""- ). To facilitate the compu- 
tation of these two forces I subjoin a table. The first 
column shews the impinging angle for every five degrees 
of the qiutdraut. The second column gives a divisor 
(L) due to the coefficient of the perpendicular force 
= sin.* Jt - ^ .-■ The remaining columns give the divisors 
(F) due to the coefHcient of the parallel force= X (con. " «n.)^ 



274 





L 

DiTtoor dii* 
to literal 


F DIVISORS DUE TO THE FRONT SDBFACB. 


Degree of 
angle of 


later. tpM*. 


hiUt.tftet. 


Inter, tpaoe. 


r.ler. iptct 


Inter, ipace. 


Incidence. 


wirfiK«. 


= 1 


"i 


= * 


= » 


= i 





iuf. 


inf. 


inf. 


inf. 


inf. 


inf. 


5 


195348 


4273 


4883 


5697 


6837 


8546 


10 


49212 


2169 


2479 


2892 


3470 


4338 


15 


22152 


1484 


1696 


1978 


2373 


2968 


20 


12684 


1154 


1319 


1538 


1846 


2308 


25 


830a 


968 


1106 


1291 


1549 


1936 


30 


5936 


856 


978 


1141 


1371 


1714 


85 


4506 


789 


902 


1052 


1352 


1690 


40 


3591 


753 


860 


1004 


1206 


1506 ■ 


45 


2968 


742 


848 


989 


1187 


1484 


50 


2529 


898 


898 


1004 


1205 


1506 


55 


2211 


1127 


1127 


1127 


1352 


1610 


60 


1978 


1484 


1484 


1484 


1484 


1714 


65 


1806 


2077 


2077 


2077 


2077 


2077 


70 


1680 


3171 


3171 


3171 


3171 


3171 


75 


1590 


5538 


5538 


5538 


5538 


5538 


80 


1530 


12303 


12303 


12303 


12303 


12303 


85 


1493 


48837 


48837 


48837 


48837 


48837 


90 


1484 


inf. 


inf. 


inf. 


inf. 


inf. 



The use of this table is as follows : 

To find the resistance against the sides of the train: 
Multiply together the square of the wind^s velocity = Y', 
and the total lateral surface = K {n + 2), and divide this 
product by the number in column L due to the impinging 
angle ; the quotient is the resistance in pounds. 

To find the resistance against the front of the train: 
Multiply together the square of the wind^s eflfective velocity 
__ Y^ + 2VY, and the front surface = S ; also divide the 
number of carriages plus one by the number in column F, 
corresponding to the impinging angle and the interme- 
diate space^ and to the quotient add .002692; then multiply 
the first product and this sum together : the result is the 
resistance in pounds^ which becomes positive or negative 



275 

dependent on the direction and the velocity of the train 
as already pointed out. 

The Resistance of Curves. 
As lon^ as the train proceeds over those portions of a 
railway which are straight^ the progressive motion is not 
disturbed by lateral forcible contact with the rails, because 
there is no disturbing force which urges the wheels to leave 
the rectilinear course in which they move in accordance to 
the laws of dynamics. But when the train arrives at such 
portions of a railway as are laid down in a curvilinear 
direction, then the flanges of the wheels, by reason of the 
dynamical law of progression, come into contact with the 
rail whose radius is greatest, so that during the train's 
progress along the curve they are constantly urged to 
impinge on the rail, which diverts the wheels jfrom their 
tendency to continue motion in a right line. This con- 
tinual contact between the flange and the rail is a source 
of resistance which I propose to investigate. It is well 
known that when a body is made to revolve round a centre, 
it is acted upon by two forces ; one urges the body forward 
in the direction of the tangent to the circular path of the 
body, and the other urges the body towards the centre of 
revolution : the combined action of these two forces keep 
the body in rotation. The latter force is equal and oppo- 
site to another which urges the body to fly from the centre 
and to seek a rectilineal path, and which is called the 
centrifugal force of the body. This force may be easily 
appreciated by observing the effect of a common sling which 
a man puts into rs^id revolution. As long as the man holds 
the strings together the sling revolves; but when he lets one 
of them sKp, the stone Or weight in the sling is liberated and 
goes forward in a straight Une. The force urging the body 
towards the centre is measured by the tension on the string, 
and this tension is equal and opposite to the centrifdgal 
force. In like manner when a train moves along a curve 
the tractive power would urge the carriages in a straight 
line were they not confined by the rail> the pressure 



276 

against the rail is the measure and effect of the centrifugal 
force, and the friction consequent on the pressure is the 
measure of the resistance to progression. The rule given 
in all works on mechanics for ascertaining the centrifugal 

force is ~ =P when Vis the velocity in feet per second 

32 >< R • . ' 

R is the radius in feet, W is the weight of the body. 
But in this work we shall require that 

V = velocity in miles per hour. 

r = radius in miles. 

W = weight of the train in tons = 5 (w + 4). 

P = centrifugal force in pounds. 

The above expression becomes P = — -^ = zUl^Lz), 

Engineers have generally attempted to balance this cen- 
trifugal force by making the exterior rail of the curve of 
such an elevation above the interior one, so that the gravity 
of the carriages will cause them to descend a transverse 
inclined plane, whose length is to its height as the whole 
weight of carriages is to such a portion of it as will balance 
the centrifugal force. In order to estimate this inclined 
plane, let ff be the gauge of the rails in feet ; and h the 
elevation of the exterior above the interior rail, in inches. 

Then as ^ : *- : : 5 (» + 4) x 2240 : ^^Ol±±). 
^ 12 ^ 7r 

■WTO 

Hence A = J^ in inches. 

6533 r 

It is necessary to fix some velocity by which the value 
of h may be determined, and, of course, the maximum 
velofeity should be assumed, as then any less velocity cannot 
produce sufficient centrifugal force to surmount the incUne. 
It is, moreover, worthy of observation, that the elevation 
of the exterior rail is idtogether independent of the weight 
drawn. The following is a small table adapted to a velocity 
of 30 miles per hour as a maximum, and a gauge of rails 
= 5 feet. The elevations corresponding to any other gauge 
miay be found by multiplying the tabular number by the 
assiuned gauge, and dividing by five. 



277 



ItadiuB of 


Height of exterior 


Radius of 


Height of exterior 


curves. 


rail above inte. rail. 


curve. 


rail above inte. rail. 


in miles. 


in inches. 


in nules. 


in inches. 


* 


2.7564 


2 


.3445 


i 


1.3782 


2i 


.2756 


i 


.9188 


3 


.2297 


1 


,6891 


34 


.1969 


1} 


.5513 


4 


.1722 


.4593 


5 


.1378 


U 


.3938 


6 


.1148 



It may^ therefore^ be assumed that the centrifugal force 
is balanced by the elevation of the outward rail, and that 
resistance will not ensue from this cause. The principal 
remaining cause of resistance is the tendency of the wheels 
to impinge against the rails, causing the carriages to oscil- 
late from side to side. This takes place in the following 
manner : — ^When the train leaves the straight line, and 
comes into the curve, it continues its direct motion till the 
wheels strike the rail very obliquely. The carriages then 
recede from this raU at an angle of recoil equal to that of 
the impingence, and continue motion in this oblique 
direction till the other wheels strike the other rail, when 
they again recede, and the same degree of oscillation is 
continually repeated. This obliquity is measured by the 
small angle contained between the rail and the wheel; and 
the angle, when greatest, is due to the arc of a circle 
whose chord is in the direction of the wheels, and which 
contains, in a portion of its length, the distance between 
the extreme points of intersection of the outer circum- 
ferences of the two sets of carriage wheels, and the upper 
surface of the rail, being also the extreme points of contact; 
and, at the end of this distance, the versed sine is the 
difference between the gauge of the rails and the gauge of 
the flanges. 

I have been particular in describing these measures, 
because fipom them only we can obtain the measure of the 
chord of impingence, as will presently be seen. 



278 

The diiFeveBiOe of the gauges varies fipom } of aa inch 
to li inches with diffirarent engineers and builders. I 
will assume a mean diiSerence of 1 inch^ which gives half 
an inch on each side between the flange and the rail when 
the train does not oscillate. It is needless to take any less 
dimension for the versed sine, because, during the repeated 
oscillations that ensue, the flange must be in contact with 
the rail on one side of the line, leaving a space = 1 inch 
between the flange and the rail on the other side. Also, 
in consequence of these oscillations, one of the &ont wheels 
will be in contact with the rail on one side, while the 
opposite beck wheel is in contact with the other rail, and 
so on ; therefore, the distance of the wheels measures a 
proportion of the chord ooljy which dqpendB for its ratio 
oa the radius of the curve. 

It is required to find the length of the chord. Let 
the distance of the wheels measure from the point of 
contact = a, and the versed sine at this distance = b ; 
let the length of the half chord = c, and the versed sine of 
the half chord = a? : the radius of the curve, as before, = r. 
Then, a, b, and r are given to find c and x. Since the arc 
in this case is very inconsiderable when compared with the 
whole circle, we will suppose that the versed sines vary 
as the squares of the chords with which they are found. 
Therefore c — a is the length of the half chord with 
which the versed sine x -^ b is found. Now, as 

(? : (c — a)« :: 07 \ x — i, or ^ = ^ « >. By the properties 
of the circle x is also = r — ^/i^^f^ = -- very nearly ; 
hence — ^^^^ = — , or %rb -=^ Zac — a 9 and 
C = ^rb^a\ then if ^= in, C =?■. I, and ^ = {am^2r)\ 

But the sine of the angle which the wheel makes with the 
rail is measured by Zx, or (^"^ + ^^) when the radius is c. 



»79 

AlsOf when the length of c is known^ the number of oscil- 
latioDs due to the impingence on the cur?e in a second 

ifill be k: Hi, when V ^ velocity in miles per hour. 

xOC 

Before we proceed to estimate the force which produces 
resistance on the curve, it is necessary to explain another 
cause of the oscillations not yet noticed. 

All the engine and carriage wheels are made of a conical 
form, tapering outwards, 'Ihe object of this form is pur- 
posely to correct the tendency, we are now discussing, of 
the wheels to strike the rails while going round a curve ; 
for the instant the wheels move laterally on the rail^ those 
on one side, in consequence of their tapering form, enlarge 
their diameters } while those on the other side are equally 
diminished ; therefore, the velocity of the wheels on the 
outar rail of the curve is greater than that of the inner 
ones ; and if the velocities of both are in proportion to the 
inn^ and outer radii of the curve respectively^ it is evident 
that there would be no tendency of the wheels to strike 
the rails, but they would always continue motion in the 
direction of the tangent to the curve. All that theory 
requires is, that the carriage wheels should have such a 
taper, that the increase of velocity in moving laterally 
one indi shall be to the whole velocity as the gauge of 
the rails is to the inner radius of the curve. If d repre- 
sents the diameter of the wheel in inches, and / the taper 
of the tire for an inch of the breadth thereof, it is evident 
that d will be proportional to the whole velocity when 

2t is the increase, therefore d : 2t :: r : a or t = ^. If 

the diametCT of the wheel is 42 inches, the gauge of the 
rails 5 feet, and the radius half a mile, we shall find 

t = ^th of inch nearly, for each inch in breadth of the 

tire. Bat> practically, this conical form does not generally 
secure the object it is designed for. The taper having 
been once determined upon, can never be varied as the 
circumstances require; for, ctBteris paribus^ the taper 



280 

ought to vary inversely as the radius of the curve, which 
it cannot, and hence arises that cause of oscillation which 
I have alluded to : but, as regards the increase of resist- 
ance fipom this cause, it will ultimately appear to be very 
insignificant, and need only be considered in extreme cases. 
I will suppose, for the sake of argument, that the taper 
is calculated for the worst, that is, the sharpest curves of 
the railway. When the wheels have enlarged their diame- 
ters by the lateral motion, the increase of velocity is ex- 
pressed by —". The mcrease of velocity proportional to 

the radius of the curve should be = ^ : therefore, as the 

r 

taper which is proper for the worst curves will give too 
great an accession of velocity to any other curve, we have 

V (— — ^) for the relative velocity which will cause thewheels 

d r 

to rotate round a centre whose radius is S . But this rota- 

tion produces a centrifugal force which impels the wheels 
already oscillating into a still more rapid state of oscilla- 
tion. The measure of this centrifugal force is expressed 

by the equation = 1505.8 / V^ x ^— =^'. This force 

in conjunction with that due to the angle of impingence, 
increases the number of oscillations computed by the chord 
c, in the proportion of the latter force to the sum of the 
two : and the chord c will be shortened in the same ratio. 
The force p, due to the angle of impingence, is to the 
whole force which propels the weight of the carriage, as 

the sine of the angle = (?!? + 2r)' to the radius = ?5L±_2r ; 

and p = 5600 x «.5±2r. The sum of the two forces becomes 

mr 

= 5600 ["■^'^jiJr + ,2689 t V x ^'^*^~^^\ consequently 

the number of oscillations will be 

= V X 1^^^ H- .7887 V X ^ X (f- ^^', and 

flwi + 2r or gr {am + 2r)' 



281 

when is known, the length of the chord of impiugence is 
22V 

I 15,0 

I In order to enable the reader to apply* these rules prac" 

ticaUy, I will assume such general dimensions as will sim- 
plify the formula, and then give a practical example in 
illustration. 

Let the distance of the wheels, as specified in a former 
page, be a = 8i feet, and the lateral play of the wheels be 

d = 1 inch, then -? = w = 100. Let the diameter of the 

o 

carriage wheels be rf = 42 inches, and the taper of the 

tire be /=.05 inches ; let the gauge of the rails be^=5 feet, 

also let V = 30 miles per hour, and let r be given in miles ; 

so in that r in the formula becomes 5280r. With these 

data we shall find that the sum of the imipinging forces is 

^^ ^ 8.8384 ■.Ii2r ^ (2.5i42r---i)' j The number of oscilla- 

^ r 3456/^ 

tionsisrepresentedbyo^, }^'^\ , + ^.^^ x (^-Si^^^D^ 

^ ^ (I2.6723r+1) r 12.6723r+l)2 

If we take, for an example, the radius to be 1 mile, then the 
force acting at the point of contact is=(120.8384 + .0006634) 
= 120.839 lbs. per carriage. 

As it is evident that the second term of both the above 
equations due to the taper of the wheels is scarcely 
appreciable except . at very great velocities and round very 
sharp curves, it is better to omit it altogether in the estimate; 
and then we have the following practical rules adapted to 
the data already given : 

Rule — To find the, force of impingence on the curve: 
Add together 112 and the quotient of 8.838 divided by 
the radius in miles : the sum is the resistance due to 
the impinging force in pounds common to aU velocities. 

Rule — To find the number of oscillations in a second of 

time : Divide the nimiber 10.56 by the sum of 1 + 12.67 times 

. the radius : the quotient is the number required. If the 

number for any other velocity is wanted, multiply this 

result by the given velocity, and divide by 30. 

G G 



282 

The oscillations of the wheels computed by the foregoing 
rule must not be confounded with the actual oscillations 
observed by a passenger travelling in a carriage^ which are 
produced by the lateral play of the springs and bearings, 
and are much more frequent. It may be easily shown 
that the engine is more liable to oscillation than the car- 
riages, from its greater weight and the inequality of the 
diameters of the wheels and conical tires, also because the 
centre of gravity of the machine is higher, and there is an 
unequal action produced by the successive reciprocating 
motion of the two crank?. But this oscillation is not 
communicated to the cstrriages, and since my intention 
has been to offer a general rule oi^ily, I shall not take any 
account of it. 

Since the resistance computed by the foregoing rule is not 
uniform but chronic, it is now proper to determine the mean 
tractive force required to overcome it. Because the ex- 
pression B600 (?^L±ir) represents the force of impingence 

of one cforiage, and v x ^•^^^"' represents the oscillations 
in a second of time, then 5600 (« + 4) x ?:?33© _. ^^^ ^tole 

T 

force of impingeujee aeting duriiig this time. But the 
lateral velocity with which this force moves is to the ffEwr- 
ward velocity as the sine of the length to the radiufi, ^ 
^ . V :: (?!?L±2tf . am f gr therefore v = V k ??L±ir ; 

4mV 2m 2mr 

consequently the force with which the whole train r§9ists 
uniformly is 5600 (« + 4) V x ii666^mj_2r) . ^^ ^j^^ 

this expression is simplified to suit the dato* already assumed, 

the rule becomes ^-^ x ^^'^7^J' f ^ = uniform maximum 

13.676 y* 

resistance. 

It must be borne in mind that this resistance js tha 
greatest which can ever occur, because, by hy^thesis. 



283 

the extreme lateral play of the wheels was assumed. But 
the carriages will oscillate though the lateral motion be 
not extreme, and it is required to know what the resistance 
will be in any given case. This may be easily accomplished 
by observing the number of oscillations made in a second 
of time. The resistance wiU vary as the squares of the 
sines of the angles of impingenoe. But the angle of 
impingence depends upon the ratio of the chord to the 
radius of the curve, and this ratio will be obtained by 
supposing that the radius varies inversely aa the number 
of oscillations, for then it is evident that the length of the 
chord being the same, the radius for a less number of 
oscillations will be greater, and consequently the angle of 
impingence less, and the effect produced will be the same 
as if the space travelled over were greater before the wheels 
came in contact with the rail; so that we have the following 
method to determine the actual resistance : 

Let O bQ the observed number of oscillations: when the 

radius = 1, the sine = - x ^^ "** ^^ due to O; consequently 

the resistance will vary as - ; hence the expression for 

the resistance with any observed number of oscillations is 
954.54 ^ X (n+4) x C«2+Hr)' ^^ ^gj^g ^^e ^^ta as before, 

(n+4) >< ^ X ^iMZH!L±ir = general expression of resistance 

in pounds. 

Rule — To find the resistance qfa curve {acting unifoimily 
to retard motion) : Multiply the number of carriages plus 4 
by the square of the oscillations observed in a second, and 
by the cube of the sum of 1 + 12.67 times the radius, and 
divide this product by 1514 times the square of the radius : 
the quotient is the resistance of the train in pounds at a 
velocity of 30 miles per hour. For any other velocity divide 
this result by the given velocity, and multiply the quotient 
by 30, the product is the resistance sought. 



284 

Example : 
Given « = 10 carriages. 

0=1 oscillation per second, 
r = 1 mile radius, 
rrten (10^4 =14) xi»x (12.67^1)3 ^ j^g g^ j^^ resistance. 

1514 

In treating of this subject^ I am aware that the informa- 
tion we possess is by no means satisfactory, and is very 
scanty j and indeed, the foregoing estimate of the resistance 
I wish the reader to accept as suggestive only. But, were 
there a few good experiments made, the calculation might 
be reduced to almost a certainty, by the use of the fore- 
going investigation. For, if the actual resistance were 
ascertained with given dimensions, then a practical rule 
can be formed by making the resistance vary directly as 
the square of the number of oscillations, the radius, and the 
number of carriages ; and inversely as the velocity. When 
the resistance is determined for any of the set of wheels, 
whose distances are respectively 90, 100, 110, 120, 130 
inches, then the resistance due to any other set will vary 
inversely as the numbers 2077, 1514, 1137, 876, 689, 
respectively, the velocity, radius of curve, number of car- 
riages, and number of oscillations per second, remaining 
the same in all cases. Also, we derive that the resistance 
due to curves is not great, except in extreme cases, and it 
need not be generally regarded when the centrifugal force 
is properly balanced. 

As it may be convenient to know what diflPerence a varia- 
tion of the distance between the wheels makes in the 
estimate, I subjoin the requisite alterations of the coeffi- 
cients employed in the foregoing rules. 



285 





When the distance is t 


[ =c 


Instead of the coefficient. 


Inches. 


Inches. 


Inches. 


Inches. 




90 


110 


120 


130 


8.838 substitute. 


7.95 


9.72 


10.60 


11.49 


12.576 do. 


15.08 


12.34 


11.31 


8.29 


12.672 do. 


15.63 


10.47 


8.80 


7.50 


10.560 do. 


11.73 


9.60 


8.80 


8.12 


1514,000 do. 


2077,00 


1137.00 


876.16 


689.18 



If it is required to know the resistance caused by oscil- 
lation on a straight line, the following form will give it : 

Resistance is = ^^ ^^ ^^) ^' in pounds. 

The literal expression of the above form may be more 
convenient, therefore I herewith subjoin it. 

To find the resistance on a straight line due to the oscilla- 
tion of the carriages : 
Multiply together, the coefficient 106, the number of 
carriages plus 4, and the square of the number of oscilla- 
tions per second ; divide the product by the velocity in 
miles per hour : the quotient is the resistance required in 
pounds. 

The friction consequent upon the resistance may be 

assumed to be ]^th of the result derived from the rules 

whether for a curve or for a straight Une. 

I shall now conclude this essay; and I beg the reader 
to consider that I do not put forth the data I have 
adopted as invariable, but have singled them out from 
the many practical cases that occur, rather for the sake of 
argument. Should the reader be desirous of adopting his 
own, or any other practical data, I shall be happy, if in 
the preceding pages I have been instrumental as a guide 
in leading his inquiries to an approximate, if not an accu- 
rate result. Though the consideration of the acceleration 
or retardation of trains is a subject which may be properly